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1. Introduction Duality is a special type of symmetry that involves with “polar opposites” and their dynamical interplays. The two (“Yin” and “Yang”, studied by a legendary sage Lao Tzu in his book Tao Te Ching and in the Book of Changes or I-Ching) are not merely opposites. The more we learn about human striving, the more we see they are supplementary, complementary, integrative and inextricably bound together. Duality is very elegant, yet powerful and has a long and distinguished history going back thousands of years. It is a natural and precious phenomenon that permeates or occurs in practically all branches of mathematics, physics, engineering, logic, psychology, real life, food science, social sciences, natural sciences, medical sciences such as alternative or energy medicine, acupuncture, meditation, qigong, physical therapy, nutrition therapy, immunotherapy, etc. Fundamentally, duality gives two different points of view of looking at the same object. In the study of comparison theorems (in differential equations, differential geometry, differential - integral inequalities), harmonic forms, mean curvature, etc, we find many objects that have two different points of view and in principle they are all dualities. Whereas, in physics electricity and magnetism are dual objects, Hodge theory of harmonic forms is motivated in part by Maxwell’s equations of unifying magnetism with electricity in a physics world. In a complex line bundle LC with structure group U(1) over a 4-dimensional Lorentzian manifold, Maxwell’s equations are the precise equations for the curvature 2-form ω of a connection on ⋆ LC being both closed (dω = 0) and co-closed (d ω = 0), and hence the curvature form ω is harmonic ∆ω := (dd⋆ + d⋆d)ω =0 . Whereas in topology, duality can broadly distinguish between−contravariant functors such as comology, K-theory, or more general bundle theory and covariant functor
such as homology or homotopy. A de Rham cohomology class involves with interplaying two objects - closed forms (coming from the source of one exterior differential operator d) and exact forms (coming from the target of its preceding operator d). Whereas in operator theory, Hodge Laplacian involves with utilizing dual objects - exterior differential operator d and codifferential operator d⋆, harmonic forms are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian. From the view point of calculus of variations, harmonic forms are precisely the minimal L2 forms within their respective topological (cohomology) classes. Whereas, many things embrace dualities Yin and Yang according to Lao Tzu, their dynamical interplay is viewed as the essence of all natural phenomena, human affairs, and Chinese medicine, and has strong impacts, interactions and connections with unity. Harmonic forms, mo- tivated in part by Maxwell’s equations, coming out of physics, generalize harmonic functions in the study of complex analysis, partial differential equations, potential theory, and have important connections with several complex variables, Lie group representation theory and algebraic geometry through the use of K¨ahler metric and Bochner technique. In the first part of this paper, we begin with studying comparison theorems in differential equations and in differential geometry and the transitions between these two fields from the viewpoint of dualities. We observe and describe the duality between two types of differential equations, the second-order, linear Jacobi type equation (2.1) and the first-order, nonlinear Riccati type equation (2.2), and the duality between the initial condition in (2.1) DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS3 and the asymptotic condition in (2.2) (see Theorem 2.1). Moreover, to each type of differential equation, there corresponds a comparison theorem on its supersolutions and subsolutions with appropriate initial or asymptotic condition (see Theorems E and Theorem 2.2). The duality between these two types of differential equations (2.1) and (2.2) leads to the duality between two corresponding comparison theo- rems in differential equations of the same type. This, in turn gives rise to new duality in swapping differential equations of dual type and hence generates the fol- lowing Comparison Theorems in Differential Equations of Mixed Types I and II with appropriate initial and asymptotic conditions: Denote AC(0,t) the set of absolutely continuous real-valued function on an open interval (0,t), i.e., AC(0,t)= f : (0,t) R f is absolutely continuous , where (0,t) (0, ) . { → | ⊂ ∞ } Theorem 2.3. (Comparing Differential Equations of Mixed Type I) Let Gi be real- valued functions defined on (0,ti) (0, ) ,i =1, 2 satisfying ⊂ ∞ (2.15) G G 2 ≤ 1 on (0,t ) (0,t ). Let g AC(0,t ) be a solution of 1 ∩ 2 1 ∈ 1 2 g1 g1 ′ + + κ1G1 0 a. e. in (0,t1) (2.24) κ1 ≤ κ1 + (g1(t)= + O(1) as t 0 t → 1 and f C([0,t ]) C (0,t ) with f ′ AC(0,t ) be a positive solution of 2 ∈ 2 ∩ 2 2 ∈ 2 f ′′ + G f 0 a. e. in (0,t ) (2.14) 2 2 2 ≥ 2 (f2(0) = 0 ,f2′(0) = κ2. with constants κi satisfying (2.16) 0 <κ κ . 1 ≤ 2 Then t t and 1 ≤ 2 κ2f2′ (2.30) g1 ≤ f2 on (0,t1) . Theorem 2.4. (Comparing Differential Equations of Mixed Type II) Let functions 1 Gi : (0,ti) (0, ) R, i =1, 2 satisfy (2.15). Let f C([0,t ]) C (0,t ) with ⊂ ∞ → 1 ∈ 1 ∩ 1 f ′ AC(0,t ) be a positive solution of 1 ∈ 1 f ′′ + G f 0 a. e. in (0,t ) (2.13) 1 1 1 ≤ 1 (f1(0) = 0 ,f1′(0) = κ1 and g AC(0,t ) be a solution of 2 ∈ 2 2 g2 g2 ′ + + κ2G2 0 a. e. in (0,t2). (2.25) κ2 ≥ κ2 + (g2(t)= + O(1) as t 0 , t → Assume (2.16). Then t t and 1 ≤ 2 κ1f1′ (2.32) g2 f1 ≤ on (0,t1) . 4 S.W. WEI

When the radial Ricci curvature of a manifold is bounded below by a func- tion (n 1) G1 cf . (2.34) , we show via Weitzenb¨ock formula (Theorem F), this is a disguised− supersolution of a Riccati type equation (2.38). Similarly, when
the radial curvature of a manifold is bounded above by a function G˜2 cf . (2.60) (resp. bounded below by a function G cf . (2.59) , we show this is a disguised sub- 1
solution resp. supersolution of a Riccati type equation (2.28) resp. (2.27) in
which G = G˜ and t = t˜ derived from (2.66) resp. (2.65) . Thus, we are ready 2 2 2 2

to utilize dualities in comparison theorems in differential equations of mixed types
to generate comparison theorems in Riemannian geometry and provide simple and direct proofs. In particular, we obtain Laplacian Comparison Theorem 2.5 when the radial Ricci curvature is bounded below as in (2.34), and Hessian and Laplacian Comparison Theorem 2.8 when the radial curvature is bounded above as in (2.60) (resp. bounded below as in (2.59)). Throughout this paper we fix a point x0 in an n-dimensional manifold M. Let r be the distance function on M relative to x0 , D(x ) = M (Cut(x ) x ) , Bt(x ) = x M : r(x) < t , and a punctured 0 \ 0 ∪{ 0} 0 { ∈ } ◦ geodesic ball Bt(x )= Bt(x ) x . 0 0 \{ 0}

Theorem 2.5. (Laplacian Comparison Theorem) Let functions Gi : (0,ti) (0, ) R, i =1, 2 satisfy (2.15) on (0,t ) (0,t ). Assume ⊂ ∞ → 1 ∩ 2 (2.34) (n 1) G (r) RicM (r) − 1 ≤ rad ◦ 1 on Bt1 (x0) D(x0), and let f2 C([0,t2]) C (0,t2) with f2′ AC(0,t2) be a positive solution⊂ of ∈ ∩ ∈

f ′′ + G f 0 a. e. in (0,t ) (2.35) 2 2 2 ≥ 2 f (0) = 0 ,f ′ (0) = n 1 . ( 2 2 − Then t t and 1 ≤ 2 f (2.36) ∆r (n 1) 2′ (r) ≤ − f2

◦ holds in Bt1 (x0) . If in addition, (2.34) occurs in D(x0) , then (2.36) holds pointwise on D(x0) and weakly on M.

Theorem 2.8. (Hessian and Laplacian Comparison Theorems) Let (2.59) G (r) K(r) 1 ≤ ◦ on Bt (x ) D(x ) resp. 1 0 ⊂ 0 (2.60) K(r) G (r) ≤ 2 ◦ 1 on Bt˜2 (x0) D(x0) , and let f2 C([0,t2]) C (0,t2) with f2′ AC(0,t2) be a positive solution⊂ of f ∈ ∩ ∈

f ′′ + G f 0 a. e. in (0,t ) (2.14) 2 2 2 ≥ 2 (f2(0) = 0 ,f2′(0) = κ2, where Gi : (0,ti) R satisfy → (2.15) G2 G1 on (0,t ) (0,t ) and 1 ≤κ . 1 ∩ 2 ≤ 2 DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS5

1 resp. f C([0, t˜ ]) C (0, t˜ ) with f ′ AC(0, t˜ ) be a positive solution of 1 ∈ 1 ∩ 1 1 ∈ 1 f ′′ + G f 0 on (0, t˜ ) (2.61) 1 1 1 ≤ 1 (f1(0) = 0 ,f1′ (0) = κ1 , f where Gi : (0, t˜i) R satisfy → (2.62) G2 G1 on (0, tf˜ ) (0, t˜ ) and 0 <κ≤ 1 . 1 ∩ 2 1 ≤ Then t1 t2 , f f ≤ κ
f ′ κ f ′ (2.63) Hess r 2 2 (g dr dr) and ∆r (n 1) 2 2 (r) ≤ f2 − ⊗ ≤ − f2 ◦ ˜ ˜ on Bt1 (x0) resp. t1 t2 , κ f ′ ≤ κ f ′ (2.64) 1 1 g dr dr Hess r and (n 1) 1 1 (r) ∆r f1 − ⊗ ≤ − f1 ≤ ◦ on Bt˜1 (x0) , in the sense of quadratic
forms. If in addition (2.59) occurs on D(x0), then the second part of (2.63) holds pointwise on D(x ) and weakly in M.
0 This strengthens main theorems in [24, Theorem 4.1], and [38, Theorem D] by weakening the hypotheses on the domain and regularity of Gi (resp .G˜i ) from a + smooth function on R 0 to an arbitrary real-valued function on (0,ti) resp.(0, t˜i) , (not necessarily continuous)∪{ } and the hypotheses on the domain of K(r) from D(x ) 0
◦ ◦ to Bt1 (x0) resp . Bt˜2 (x0) . We also give direct and simple proofs with new appli- cations. In applying Theorem
2.5 under the radial Ricci curvature assumption, (2.39) A(A 1) M G = (n 1) − Ric , A 1, or equivalently (2.41), we need (2.35) with 1 − − r2 ≤ rad ≥ appropriate G2 for comparison. But in Theorem 2.6 we do not need to assume (2.35) as in Theorem 2.5. Instead, utilizing G1, we find (2.44) as a companion f ′ system for comparison by duality, and estimate 2 (see Theorem 2.6 and its proof). f2 Similarly, without assuming (2.35), we have

Theorem 2.7. (1) If B1(1 B1) M (2.51) (n 1) − Ric (r), where 0 B 1 − r2 ≤ rad ≤ 1 ≤ ◦ on Bt (x ) D(x ), then 1 0 ⊂ 0 1+√1+4B1(1 B1) (2.52) ∆r (n 1) − ≤ − 2r ◦ in Bt1 (x0). If in addition, (2.51) occurs on D(x0) , then (2.52) holds pointwise on D(x0) and weakly on M. (2) Equivalently, if B1(1 B1) M (2.53) (n 1) − Ric (r) , 0 B 1 − (c+r)2 ≤ rad ≤ 1 ≤ ◦ ◦ on Bt (x ) D(x ) , where c 0 , then (2.52) holds in Bt (x ). If in addition 1 0 ⊂ 0 ≥ 1 0 (2.53) occurs on D(x0), then (2.52) holds pointwise on D(x0) and weakly on M.

As applications of Theorem 2.8, we have Theorems 3.1 and 3.2 under the negative A(A 1) lower bound of the radial curvature assumption (3.1) − K(r) , A 1 or − r2 ≤ ≥ equivalently (3.4) , and under the negative upper bound of the radial curvature A (A 1) assumption (3.5) K(r) 1 1− , A 1 or equivalently (3.7) respectively.
≤− r2 1 ≥
6 S.W. WEI

Corollary 3.1. (1) If the radial curvature K satisfies (3.14) A(A 1) A (A 1) − K(r) 1 1 − on M x where A A 1 , − r2 ≤ ≤− r2 \{ 0} ≥ 1 ≥ then we have (3.15) A1 g dr dr Hess r A g dr dr in the sense of quadratic forms, r − ⊗ ≤ ≤ r − ⊗  A1  A  (n 1) r ∆r (n 1) r pointwise on M x0 and − ≤ A ≤ − \{ } ∆r (n 1) r weakly on M. (2) Equivalently, if≤K satisfies− A(A 1) A (A 1) (3.16) − K(r) 1 1− , A A 1 , − (c+r)2 ≤ ≤− (c+r)2 ≥ 1 ≥ on M x where c 0 , then (3.15) holds. \{ 0} ≥ Corollary 3.1 is equivalent to the following Theorem A in which (1) is due to Han-Li-Ren-Wei ([24]), and extends the work of Greene and Wu ([21, p.38-39]) from asymptotical estimates off a geodesic ball B(a 1)(x0) near infinity to pointwise estimates on M x − \{ 0}
Theorem A. (1)([24]) Let the radial curvature K satisfy A A (3.17) K(r) 1 where 0 A A − r2 ≤ ≤− r2 ≤ 1 ≤ on M x0 . Then (3.18)\{ } 1+ √1+4A 1+ √1+4A 1 g dr dr Hess(r) g dr dr on M x , 2r − ⊗ ≤ ≤ 2r − ⊗ \{ 0}     (3.19) 1+ √1+4A 1+ √1+4A (n 1) 1 ∆r (n 1) pointwise on M x , and − 2r ≤ ≤ − 2r \{ 0}

1+ √1+4A (3.20) ∆r (n 1) weakly on M. ≤ − 2r (2) Equivalently, if A A (3.21) K(r) 1 where 0 A A − (c + r)2 ≤ ≤−(c + r)2 ≤ 1 ≤ on M x , then (3.18)-(3.20) hold. \{ 0} As further applications of Theorem 2.8, we also obtain the following Theorems 3.3 and 3.4 under positive lower and upper bound on radial curvature assumptions respectively. Theorem 3.3. (1) If B (1 B ) (3.25) 1 − 1 K(r) where 0 B 1 r2 ≤ ≤ 1 ≤ ◦ on Bt (x ) D(x ), then Hess r and ∆r satisfy 1 0 ⊂ 0 1+√1+4B1(1 B1) (3.26) Hess r − g dr dr and ≤ 2r − ⊗   1+√1+4B1(1 B1) (3.27) ∆r (n 1) − ≤ − 2r DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS7

◦ on Bt1 (x0) respectively. If in addition (3.25) occurs on D(x0) , then (3.26) holds pointwise on D(x0) and (3.27) holds weakly on M. (2) Equivalently, if B (1 B ) (3.28) 1 − 1 K(r) , 0 B 1 (c+r)2 ≤ ≤ 1 ≤ ◦ ◦ on Bt (x ) D(x ), where, c 0 , then (3.26) and (3.27) hold on Bt (x ). If in 1 0 ⊂ 0 ≥ 1 0 addition (3.28) occurs on D(x0), then (3.26) holds pointwise on D(x0) and (3.27) holds weakly on M. Theorem 3.4. (1) If B(1 B) (3.35) K(r) − where 0 B 1 ≤ r2 ≤ ≤ ◦ on Bt (x ) D(x ), then Hess r and ∆r satisfy 2 0 ⊂ 0 B 1 + 1 (3.36) | − 2 | 2 g dr dr Hess r and r − ⊗ ≤ B 1 + 1  (n 1) | − 2 | 2 ∆r − r ≤ ◦ on Bt2 (x0) respectively. (2) Equivalently, if B(1 B) (3.37) K(r) − , 0 B 1 ≤ (c+r)2 ≤ ≤ ◦ ◦ on Bt (x ) D(x ) , where c 0 , then (3.36) holds on Bt (x ). 2 0 ⊂ 0 ≥ 2 0 Theorem 3.5. (1) If the radial curvature K satisfies B B 1 (3.45) 1 K(r) where 0 B B r2 ≤ ≤ r2 ≤ 1 ≤ ≤ 4

◦ on Bτ (x ) D(x ) , then 0 ⊂ 0 1+√1 4B 1+√1+4B1 (3.46) − g dr dr Hess r g dr dr and 2r − ⊗ ≤ ≤ 2r − ⊗     1+√1 4B 1+√1+4B1 (n 1) − ∆r (n 1) − 2r ≤ ≤ − 2r ◦ hold pointwise on Bτ (x0). If in addition (3.33) occurs on D(x0) , then 1+ √1+4B (3.47) ∆r (n 1) 1 ≤ − 2r holds weakly on M. (2) Equivalently, if K satisfies B B 1 (3.48) 1 K(r) , 0 B B (c + r)2 ≤ ≤ (c + r)2 ≤ 1 ≤ ≤ 4

◦ ◦ on Bτ (x ) D(x ) , where c 0 , then (3.46) holds on Bτ (x ). If in addition, 0 ⊂ 0 ≥ 0 (3.33) occurs on D(x0) , then (3.47) holds weakly on M. B 1 The case K(r) r2 , B 4 is due to Han-Li-Ren-Wei ([24]). Theorem 3.5 is equivalent to the following≤ ≤ Corollary 3.5. (1) Let the radial curvature K satisfy B (1 B ) B(1 B) (3.49) 1 − 1 K(r) − where 0 B,B 1 r2 ≤ ≤ r2 ≤ 1 ≤

◦ on Bτ (x ) D(x ) . Then 0 ⊂ 0 8 S.W. WEI

1 1 B + 1+√1+4B1(1 B1) (3.50) | − 2 | 2 g dr dr Hess r − g dr dr and r − ⊗ ≤ ≤ 2r − ⊗  1 1    B + 1+√1+4B1(1 B1) (n 1) | − 2 | 2 ∆r (n 1) − − r ≤ ≤ − 2r ◦ hold pointwise on Bτ (x0). If in addition (3.25) occurs on D(x0) , then 1+√1+4B1(1 B1) − (3.27) ∆r (n 1) 2r holds weakly on M. ≤ − (2) Equivalently, if K satisfies B (1 B ) B(1 B) (3.51) 1 − 1 K(r) − , 0 B,B 1 (c+r)2 ≤ ≤ (c+r)2 ≤ 1 ≤ ◦ ◦ on Bτ (x ) D(x ) , where c 0 , then (3.50) holds on Bτ (x ). If in addition, 0 ⊂ 0 ≥ 0 (3.28) occurs on D(x0) , then (3.27) holds weakly on M. Corollary 3.7. If the radial curvature K satisfies A B K(r) −r2 ≤ ≤ r2 (3.54) A B 1 (resp. K(r) ) , 0 A, 0 B −(c + r)2 ≤ ≤ (c + r)2 ≤ ≤ ≤ 4 on M x , where c> 0 , then \{ 0} 1+ √1 4B 1+ √1+4A (3.55) − g dr dr Hess(r) g dr dr and 2r − ⊗ ≤ ≤ 2r − ⊗     1+ √1 4B 1+ √1+4A (3.56) (n 1) − ∆r (n 1) − 2r ≤ ≤ − 2r hold on M x . \{ 0} As immediate applications of Comparison Theorems in Differential Geometry (Theorems 3.1-3.5), we obtain Mean Curvature Comparison Theorems (see Theo- rems 4.1). We then utilize dualities to study harmonic forms and their decompostion, in- tegrability and growth. It is well-known that on a compact Riemannian manifold, a smooth differential form ω is harmonic if and only if it is closed and co-closed. That is, (6.1) ∆ω = 0 if and only if dω = 0 and d⋆ω =0. On complete noncompact Riemannian manifolds, although (6.1) holds for smooth ω with compact support, it does not hold in general. Simple examples include, in n R , a closed, non-co-closed, harmonic form ω1 = x1dx1 , a non-closed, co-closed, harmonic form ω2 = xndx1 , and a non-closed, non-co-closed, harmonic form ω3 = (x1 + xn)dx1 , or ω4 = x1xndx1 + xndxn (see [39]). However, it is proved in [3]. Theorem B. (A. Andreotti and E. Vesentini [3]) On a complete noncompact Rie- mannian manifold M, (6.1) holds for every smooth L2 differential form ω . It is interesting to explore any possible generalizations of Theorem B, in partic- ular, to discuss whether or not (6.1) holds for Lq differential form ω , where q =2 . Some study of this generalization can be found in [43, p.663, Proposition 1],6 its counter-examples are given by D. Alexandru-Rugina (see [1, p. 81, Remarque 3]) m on the hyperbolic space H 1,m 3 , and relevant remarks of Pigola-Rigoli-Setti are discussed in [31, p.260,− Remark≥ B.8]. DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS9

In [39], we introduce and add Condition W see (6.5), where Ω = ω and Ak(ξ)= Ak to the above work, so that the counter-examples provided in [1] cannot happen, the conclusion of the Proposition in [43] still holds, and works in a more general setting
with geometric and physical applications. We extend the differential form in L2 space in Theorem B in several ways. To this end, recall in extending functions in L2 space (resp. in Lq space), we introduce and study the notion of function growth: “p-finite, p-mild, p-obtuse, p-moderate, and p-small” growth for q = 2 (resp. for the same value of q) , and their counter-parts “p-infinite, p-severe, p-acute, p- immoderate, and p-large” growth [40] (see Definition 5.1). We also introduce the notion of p-balanced and
p-imbalanced growth for functions and differential forms on complete noncompact Riemannian manifold M in [36]. Namely, a function or a differential form f has p-balanced growth (or, simply, is p-balanced) if f has one of the following: p-finite, p-mild, p-obtuse, p-moderate, or p-small growth on M, and has p-imbalanced growth ( or, simply, is p-imbalanced) otherwise (see Definition 5.2). Furthermore, extending our techniques for function growth (see [9, 10]) to differential form growth, we use direct simple new methods. Let Ak be the space of smooth differential k-forms on M, and ω be a smooth differential k-form, k 0 on k n k ≥ M. Denote by , and ⋆ : A A − , the inner product and the linear operator which assigns toh· ·i each k-form→ on M an (n k)-form and which satisfies ⋆⋆ = ( 1)nk+k+1 respectively. In particular, we have− the following theorem, generalizing [44]− from functions to differential forms (See also [40] for generalizations to various types of functions). Theorem C. (Unity Theorem) ([39, Theorem 4.1]) If a differential form ω Ak has 2-balanced growth, for q = 2 , or for 1 < q(= 2) < 3 with ω satisfying∈ Condition W, then the following six statements: (i) ω,6 ∆ω 0 . (ii) ∆ω =0 . (iii) dω = d⋆ω =0 . (iv) ⋆ ω, ∆ ⋆ ω 0 . (v) ∆ ⋆ ω =0h. (vi) i≥d⋆ ω = d⋆ ⋆ ω =0 . are equivalent. h i≥ In the second part of the paper, we extend the notion of function growth and differential form growth (see [36, 39]) to bundle-valued differential form growth of various types, and prove their interrelationship. In particular, we show Theorem 5.4. For a given q R , a function, or differential form or bundle-valued differential form f is ∈ p moderate (5.4) p small (5.6) p mild (5.2) p obtuse (5.3) − ⇔ − ⇒ − ⇒ − or equiavalently, p acute p severe p large p immoderate. − ⇒ − ⇒ − ⇔ − Hence, for a given q R , f is ∈ p balanced either p finite (5.1) or p obtuse (5.3) − ⇒ − − p imbalanced both p infinite and p immoderate . − ⇒ − − If in addition, f qdv is convex in r, then the following four types of growth B(x0;r) | | are all equivalent: f is p-mild, p-obtuse, p-moderate, and p-small (resp. p-severe, R p-acute, p-immoderate, and p-large), i.e., f is (5.2) (5.3) (5.4) (5.6) for the same value of q R . ⇔ ⇔ ⇔ ∈ In particular, we have 10 S.W. WEI

Corollary 5.1. Every Lq function or differential form or bundle-valued differential form f on M has p-balanced growth, p 0 , and in fact, has p-finite, p-mild, p- obtuse, p-moderate, and p-small growth,≥p 0 , for the same value of q ≥ We also introduce Condition W for bundle-valued differential forms in Ak(ξ) and extend the unity theorem from generalized harmonic forms in Ak to generalized harmonic forms in Ak(ξ) with values in a vector bundle, where ξ : E M be a smooth Riemannian vector bundle over (M,g) , i.e. a vector bundle such→ that k at each fiber is equipped with a positive inner product , E . Set A (ξ) = k h i Γ(Λ T ∗M E) the space of smooth k-forms on M with values in the vector bundle ⊗ k k+1 E ξ : E M. Let d∇ : A (ξ) A (ξ) relative to the connection be the → → k+1 k ∇ exterior differential operator, δ∇ : A (ξ) A (ξ) relative to the connection E → k k be the codifferential operator, and ∆∇ = (d∇δ∇ + δ∇d∇): A (ξ) A (ξ). Note∇ that if E is the trivial bundle M R equipped− with the canonical→ metric, k k × ⋆ then A is isometric to A (ξ) , d = d∇ as in (6.2) , d = δ∇ as in (6.3) , and k k ⋆ ⋆ k k ∆∇ : A (ξ) A (ξ) coincides with ∆ = (dd + d d): A A . Denote by → k − n
k →
the same notations , , , and ⋆ : A (ξ) A − (ξ) , the inner product, the norm induced in fibersh· ·i of various| · | tensor bundles→ by the metric of M, and the linear k n k operator which assigns to each k-form in A (ξ) on M an (n k)-form in A − (ξ) and which satisfies ⋆⋆ = ( 1)nk+k+1 respectively. − − Theorem 6.2. (Unity Theorem) If a bundle-valued differential k-form Ω Ak(ξ) has 2-balanced growth, for q = 2 , or for 1 < q(= 2) < 3 with Ω satisfying∈ Condition W (6.5), then the following six statements6 are equivalent. (i) Ω, ∆∇Ω 0 ( i.e. Ω is a generalized bundle-valued harmonic form). h i≥ (ii) ∆∇Ω=0( i.e. Ω is a bundle-valued harmonic form). ∇ (iii) d∇ Ω= δ Ω=0( i.e. Ω is closed and co-closed). (iv) ⋆ Ω, ∆∇ ⋆ Ω 0 ( i.e. ⋆ Ω is a generalized bundle-valued harmonic form). h i≥ (v) ∆∇ ⋆ Ω=0 ( i.e. ⋆ Ω is a bundle-valued harmonic form). (vi) d∇ ⋆ Ω= δ∇ ⋆ Ω=0( i.e. ⋆ Ω is closed and co-closed). This generalizes the work of Andreotti and Vesentini [3] for the case Ω Ak and Ω is L2 , Theorem C ([39, Theorem 4.1]) , and leads to its immediate extension,∈ Corollary 6.1. Let Ω Ak(ξ) be in L2, or in Lq, 1 < q(= 2) < 3 satisfying Condition W see (6.5) .∈Then Ω is harmonic if and only if 6 Ω is closed and co- closed.
In [41], using an analog of Bochner’s method or “B2 4AC 0” method, Wei and Li derive a geometric differential-integral inequality− on a≤ manifold (see Theorem 7.1) and give the first application of the Hessian comparison theorems to generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds(see Theorem 7.2) . The case M = Rn is due to L. Caffarelli et al. [7] and is sharp due to Costa [14]. Furthermore, Wei and Li [41, Theorems 1 and 2, Corollaries 1.1-1.5] also gave the first application of the Hessian comparison theorems to generalized Hardy type inequalities on Riemannian manifolds. The case M = Rn is sharp (see [30, 26]). In the third part of the paper, using and extending the work in [41] and [38], we apply Laplacian Comparison Theorems 2.6 and 2.7, based on Theorem 2.5 to study geometric differential-integral inequalities on manifolds. In particular, we obtain DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS11

Theorem 7.3. (Generalized sharp Caffarelli-Kohn-Nirenberg type inequalities un- der radial Ricci curvature assumption) Let M be an n-manifold with a pole such M that the radial Ricci curvature Ricrad(r) of M satisfies one of the five conditions in 1,2 (7.5) where 0 B1 1 A are constants. Then for every u W0 (M x0 ) and a,b R , (7.3)≤ holds≤ ≤ in which the constant C is given by (7.6)∈. \{ } ∈ When A =1 , the result is sharp and we recapture theorems of Wei and Li (see [41, Theorems 3 and 4] or Theorem 7.2). Analogously, we apply Hessian Com- parison Theorems 3.1, 3.2, 3.3, and 3.4, based on Theorem 2.8 to study geometric differential-integral inequalities. In particular we obtain the following Theorem 7.4. When B1 = 0 , we recapture theorems of Wei and Li (see [41, Theorems 3 and 4] or Theorem 7.2). The result is sharp when M = Rn . Theorem 7.4. (Generalized sharp Caffarelli-Kohn-Nirenberg type inequalities un- der radial curvature assumption) Let M be an n-manifold with a pole and the radial curvature K(r) of M satisfy one of the twelve conditions in (7.7) where 1,2 R 0 B1 1 A are constants. Then for every u W0 (M x0 ) and a,b , (7.3)≤ holds≤ in≤ which the constant C is given by (7.8)∈ . \{ } ∈ Hessian comparison theorems lead to embedding theorems for weighted Sobolev spaces of functions on Riemannian manifolds (see Theorem 7.5) and various geo- metric integral inequalities on manifolds see (7.13), (7.15), (7.17), (7.19), (7.21), (7.23), (7.25) in Theorem 7.6 . Concurrently, Laplacian comparison Theorems lead to the following theorem.
Theorem 7.7. (Generalized sharp Hardy type inequality) Let M be an n-manifold with a pole satisfying M A(A 1) (2.39) Ricrad(r) (n 1) r2− where A 1 . 1,p u ≥−p − ≥ Then for every u W0 (M), r L (M) with p> (n 1)A +1, we have ∈ p 1 (n 1)∈A p u p − p (7.27) − − − | | dv u dv . p M rp ≤ M |∇ | The case A = 1 is sharp and is due
R to Chen-Li-WeiR (see [13, Theorem 5]). Fur- u p thermore, the assumption r L (M) cannot be dropped, or a counter-example is constructed in [13, in Section∈ 5]. Moreover, this theorem does not require 1,p u W0 (M x0 ) and hence even for the case p = 2 , the result is stronger than∈ Hardy’s\{ inequality} (7.25), obtained by setting a = 1 and b = 0 in Theorems 7.3 and 7.4 see Theorem 7.6(vii) . More recently, through the study of compar- ison theorems in Finsler geometry, some generalized Hardy type inequalities and generalized Caffarelli-Kohn-Nirenberg
type inequalities on Riemannian manifolds have been extended to Finsler manifolds by Wei and Wu (see [42]). In the forth part of the paper, following the framework in [38], employing and augmenting the work of Dong and Wei [17], Wei [38], we apply comparison theorems to the study of differential forms of degree k with values in vector bundles. In particular, we obtain a monotonicity formulae for vector bundle valued differential k-forms, if curvature satisfies one of the seven conditions in (8.12) (see Theorem 8.1). Whereas a “microscopic” approach to monotonicity formulae leads to celebrated blow-up techniques due to de-Giorgi ([15]) and Fleming ([19]), and regularity theory in geometric measure theory (see [18, 2, 33, 32, 25, 29]). Examining a macroscopic viewpoint of the above monotonicity formula, we derive vanishing theorems for 12 S.W. WEI vector bundle valued k-forms (see Theorem 9.1). This macroscopic viewpoint also leads to the study of global rigidity phenomena in locally conformal flat manifolds (see Dong-Lin-Wei [16]). In [17], Dong and Wei introduced F -Yang-Mills fields (When F is the identity map, they are Yang-Mills fields) (see also [20]). As an application of Theorem 9.1, we obtain Vanishing Theorem for F -Yang-Mills fields (see Theorem 9.2). A natural link to unity (see [37]) leads to Liouville theorems for F -harmonic p 1 2 maps (When F (t) = p (2t) ,p> 1 , they become p-harmonic maps, or harmonic maps if p =2 .) (see Theorem 10.1). In contrast to the work of Chang-Chen-Wei ([8]) on Liouville properties for a p-harmonic morphism or a p-harmonic function on a manifold that supports a weighted Poincar´einequality, we have Liouville theorems for p-harmonic maps (see Theorem 10.2). The techniques can be further applied to generalized Yang-Mills-Born-Infeld fields on manifolds. In particular we have vanishing theorems for generalized Yang- Mills-Born-Infeld fields (with the plus sign) on Riemannian manifolds (see Theorem 11.2). In [35], we solve the Dirichlet problem for p-harmonic maps to which the solution is due to Hamilton [23] in the case p = 2 and RiemN 0: ≤ Theorem D. ([35]) Let M be a compact Riemannian n-manifold with boundary ∂M and N be a compact Riemannian manifold with a contractible universal cover N. Assume that N has no non-trivial p-minimizing tangent map of Rℓ for ℓ n. Then any u Lip(∂M,N) C0(M,N) of finite p-energy can be deformed≤ to a ∈ ∩ pe-harmonic map u C1,α(M ∂M,N) Cα(M,N) minimizing p-energy in the 0 ∈ \ ∩ homotopic class with u0 ∂M = u ∂M , where 1

2. Dualities in Comparison Theory We begin with a fundamental duality in comparison theory - the duality between the second-order linear Jacobi type equation with the initial conditions at 0 and the first-order nonlinear Riccati type equation with the asymptotic condition at 0 :

2.1. Dualities between Differential Equations and Differential Inequali- ties. DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS13

Theorem 2.1. Let G(t) be a real-valued function defined on (0, τ) (0, ) , f 1 ⊂ ∞ ∈ C([0, τ]) C (0, τ) ,f > 0 in (0, τ) , f ′ AC(0, τ) ,κ> 0 be a constant and g AC(0∩, τ). Then ∈ ∈ f ′′(t)+ G(t) f(t) = 0(resp. 0 , 0) a. e. in (0, τ) (2.1) ≤ ≥ (f(0) = 0 ,f ′(0) = κ is dual to (or if and only if )

g(t)2 g ′(t)+ κ + κG(t) = 0(resp. 0 , 0) a. e. in (0, τ) (2.2) κ ≤ ≥ (g(t)= + O(1) as t 0+ . t → Furthermore, (2.1) is transformed into (2.2), via the transformer κf (2.3) g = ′ f in (0, τ) . Conversely, (2.1) is reversed from (2.2) and solution (resp. supersolution, subsolution) f(t) of (2.1) enjoys f (t) g(t) (2.4) ′ = f(t) κ in (0, τ) , via the reverser t g(s) 1 (2.5) f(t)= κt exp ds κ − s  Z0  in (0, τ). Proof. ( ) Since f > 0 in (0, τ), (2.3) is well-defined in (0, τ) with g AC(0, τ). ⇒ ∈ It follows from (2.3), the quotient law and (2.1) that 2 κf ′′f κ(f ′) g ′ = − f 2 ′ 2 2 κf (2.6) κGf f = (resp. , )− 2 ≤ ≥ f − κ
g2 = (resp. , ) κG a. e. in (0, τ). ≤ ≥ − − κ Thus, we obtain the first-order nonlinear differential equation (resp. differential inequalities ) g2 (2.7) g ′ + + κG = 0(resp. 0 , 0) κ ≤ ≥ a.e. in (0, τ) , with the initial condition at 0 in (2.1) being converted to the asymp- totic condition at 0 in (2.2). Indeed,

f (t) κ + O(t) κ 1+ O(t) g(t)= κ ′ = κ = f(t) κt + O(t2) t · 1+ O(t) (2.8) κ 1 κ = + O(t) = + O(1) as t 0+ t 1+ O(t) t →   14 S.W. WEI

( ) Observe g(s) 1 is locally bounded in a neighborhood of 0 and integrable in ⇐ κ − s [0, τ). In view of (2.5), f C([0, τ]) C1(0, τ) ,f(0) = 0 ,f > 0 on (0, τ) and ∈ ∩ f ′ AC(0, τ) . Indeed, for every t (0, τ) , ∈ ∈ t g(s) 1 t g(s) 1 g(t) 1 f ′(t)= κ exp ds + κt exp ds κ − s κ − s · κ − t  Z0   Z0    t g(s) 1 (2.9) = g(t) t exp ds · κ − s  Z0  f(t) = g(t) . · κ

+ κ t g(s) 1 Thus, (2.4) holds and as t 0 ,f ′(t)= + O(1) t exp ds κ . → t · 0 κ − s →     Hence, f ′(0) = κ , and it follows from (2.9) and (2.2) that R f(t) g(t) 2 f(t) f ′′(t)= g ′(t) + = (resp. , ) G(t)f(t) a. e. in (0, τ). · κ κ · κ ≤ ≥ − Consequently, we obtain the second-order linear differential equation (resp. differ- ential inequalities )

(2.10) f ′′(t)+ G(t) f(t) = 0(resp. 0 , 0) ≤ ≥ a.e. in (0, τ), with f(0) = 0 and f ′(0) = κ. 

Corollary 2.1. (1) Let G : (0, τ) (0, ) R, f C1(0, τ) ,f > 0 in (0, τ) , ⊂ ∞ → ∈ f ′ AC(0, τ) , and g AC(0, τ). Then ∈ ∈

f ′′(t)+ G(t) f(t) = 0(resp. 0 , 0) a. e. in (0, τ) (2.11) ≤ ≥ (f(0) = 0 ,f ′(0) = 1 is dual to (or if and only if )

2 g ′(t)+ g(t) + G(t) = 0(resp. 0 , 0) a. e. in (0, τ) (2.12) ≤ ≥ g(t)= 1 + O(1) as t 0+ . ( t → Furthermore, (2.11) is transformed into (2.12), via the transformer (2.3) in which κ = 1. Conversely, (2.11) is reversed from (2.12) via the reverser (2.5) in which κ =1, and f(t) in (2.11) enjoys (2.4) in which κ =1, (2) Theorem 2.1 is equivalent to (1). (3) Let f˜ = κf and g˜ = κg. Then the following four statements are equivalent:

f satisfies (2.11). f˜ satisfies (2.1). g˜ satisfies (2.2). g satisfies (2.12). ⇔ ⇔ ⇔

Proof. (1) Choose κ = 1 in Theorem 2.1. (2) and (3) Let f˜ = κf. Then f satisfies ˜ g˜ (2.11) if and only if f satisfies (2.1). Let g = κ . Then g satisfies (2.12) if and only ifg ˜ satisfies (2.2). Furthermore, by Theorem 2.1, f˜ satisfies (2.1) if and only if g˜ satisfies (2.2). Consequently, “(2.11) is dual to (2.12)” is equivalent to “(2.1) is dual to (2.2)” and we have shown (3).  DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS15

2.2. Dualities between Comparison Theorems in Differential Equations. To each differential equation (2.1) resp. (2.2) , there is associated Comparison Theorem E (resp. Theorem 2.2) between its supersolution and subsolution with
appropriate initial condition (resp. asymptotic condition). We state and prove these “opposites” as well as “unity” in a general form.

Theorem E. (Sturn-Liouville Type Comparison Theorem) Let Gi : (0,ti) (0, ) ⊂ 1 ∞ R be functions satisfying (2.15) on (0,t ) (0,t ), and fi C([0,ti]) C (0,ti) → 1 ∩ 2 ∈ ∩ with f ′ AC(0,ti) for i =1, 2 be solutions of the problems i ∈ f ′′ + G f 0 a . e . in (0,t ) (2.13) 1 1 1 ≤ 1 (f1(0) = 0 ,f1′(0) = κ1

f ′′ + G f 0 a . e . in (0,t ) (2.14) 2 2 2 ≥ 2 (f2(0) = 0 ,f2′(0) = κ2 with (2.15) G G 2 ≤ 1 on (0,t ) (0,t ) and 1 ∩ 2 (2.16) 0 <κ κ , 1 ≤ 2 where fi > 0 on (0,ti) ,i =1, 2 . Then

f1′ f2′ (2.17) f2 > 0 on(0,t1), in (0,t1) and f1 f2 in (0,t1). f1 ≤ f2 ≤ Proof. By (2.13), (2.14) and (2.15),

(2.18) (f ′f f ′f )(0) = 0 2 1 − 1 2 and

(2.19) (f ′f f ′f ) ′ = f ′′f f ′′f (G G )f f 0 2 1 − 1 2 2 1 − 1 2 ≥ 1 − 2 1 2 ≥ a. e. in (0,t ) (0,t ) hold. Whence f ′f f ′f 0 and 1 ∩ 2 2 1 − 1 2 ≥ f f (2.20) 1′ 2′ f1 ≤ f2 in (0,t1) (0,t2) . Integrating (2.20) from ǫ(> 0) to r(< min t1,t2 ) , and passing ǫ to 0 from∩ the right, one has { } r r ln f (t) ln f (t) and 1 t=ǫ ≤ 2 t=ǫ f1 (ǫ) κ1 f1(r) = lim f1(r) lim f2(r)= f2(r) f2(r) in [0,t1) [0,t2) . ǫ 0+ ≤ ǫ 0+ f2(ǫ) κ2 ≤ ∩ → → By the continuity of f1 and f2 , (2.21) f f 1 ≤ 2 on [0,t1] [0,t2] . Let t2 = sup t : f2 > 0 on (0,t) . We claim t1 t2. Otherwise, t < t would∩ lead to, via (2.21){ 0 < f (t ) f (t}), contradicting≤ by continuity 2 1 1 2 ≤ 2 2 f2(t2)=0 .  16 S.W. WEI

Corollary 2.2. (1) Let Gi : (0,ti) (0, ) R be functions satisfying (2.15) on ⊂ ∞1 → (0,t1) (0,t2), and let fi C([0,ti]) C (0,ti) , with fi′ AC(0,ti) be positive solutions∩ of the problems ∈ ∩ ∈

f ′′ + G f 0 a . e . in (0,t ) (2.22) 1 1 1 ≤ 1 (f1(0) = 0 ,f1′(0) = 1

f ′′ + G f 0 a . e . in (0,t ) (2.23) 2 2 2 ≥ 2 (f2(0) = 0 ,f2′(0) = 1 Then (2.17) holds. (2) Theorem E is equivalent to (1).

Proof. (1) Choose κ1 = κ2 = 1 in Theorem E. (2) Let f˜i = κifi. Then f1 satisfies (2.22) if and only if f˜1 satisfies (2.13), f2 satisfies (2.23) if and only if f˜2 ˜′ ˜′ f1 f2 satisfies (2.14). Furthermore, (2.20) holds in (0,t1) (0,t2) if and only if ˜ ˜ ∩ f1 ≤ f2 in (0,t ) (0,t ) . These imply that (2.21) holds on [0,t ] [0,t ] if and only if 1 ∩ 2 1 ∩ 2 f˜1 f˜2 in on [0,t1] [0,t2] . It follows that (2.17) holds. Consequently, Theorem E is equivalent≤ to (1).∩  Theorem 2.2. (Comparison Theorem for supersolutions and subsolutions of Ric- cati type equations) Let functions Gi : (0,ti) (0, ) R satisfy (2.15) on ⊂ ∞ → (0,t ) (0,t ), and let gi AC(0,ti) ,i =1, 2 be solutions of 1 ∩ 2 ∈ 2 g1 g1 ′ + + κ1G1 0 a. e. in (0,t1) (2.24) κ1 ≤ κ1 + (g1(t)= + O(1) as t 0 t → 2 g2 g2 ′ + + κ2G2 0 a. e. in (0,t2) (2.25) κ2 ≥ κ2 + (g2(t)= + O(1) as t 0 t → with constants κi satisfying (2.16). Then in (0,t1) , (2.26) g g . 1 ≤ 2

Proof. Proceeding as in the proof of Theorem 2.1, we use (2.5) in which f = fi, g = gi and κ = κi,i = 1, 2 , then the systems (2.24) and (2.25) are transformed into (2.13) and (2.14) respectively, satisfying (2.15) on (0,t ) (0,t ), (2.16), and 1 ∩ 2 enjoying (2.4) (in which f = fi, g = gi and κ = κi), fi > 0 on (0,ti) , fi 0 1 ∈ C ([0,ti]) C (0,ti) and fi′ AC(0,ti), i = 1, 2. It follows from Theorem E that ∩ g ∈g (2.17) holds. By (2.4), 1 2 . Consequently, via (2.16) we obtain the desired κ1 ≤ κ2 (2.26). 

Corollary 2.3. (1) Let functions Gi : (0,ti) (0, ) R satisfy (2.15) on ⊂ ∞ → (0,t ) (0,t ), and gi AC(0,ti) ,i =1, 2 be solutions of 1 ∩ 2 ∈ 2 g ′ + g + G 0 a. e. in (0,t ) (2.27) 1 1 1 ≤ 1 g (t)= 1 + O(1) as t 0+ ( 1 t → 2 g ′ + g + G 0 a. e. in (0,t ) (2.28) 2 2 2 ≥ 2 g (t)= 1 + O(1) as t 0+ , ( 2 t → DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS17

Then (2.26) holds on (0,t1). (2) Theorem 2.2 is equivalent to (1).

Proof. (1) Choose κ1 = κ2 = 1 in Theorem 2.2. (2) Enough to show that (1) Theorem 2.2. Letg ˜ = gi . Theng ˜ satisfies (2.27) if and only if g satisfies (2.24)⇒ i κi 1 1 andg ˜2 satisfies (2.28) if and only if g2 satisfies (2.25). It follows from (1) that g˜ g˜ . Hence, 1 ≤ 2 g1 g2 (2.29) =g ˜1 g˜2 = κ1 ≤ κ2 in (0,t1). Consequently, using (2.16) yields the desired (2.26) on (0,t1). 

2.3. Dualities in “Swapping” Comparison Theorems in Differential Equa- tions. We note the duality between Jacobi type equation (2.1) and Riccati type equation (2.2) leads to the duality between the above two types of correspond- ing comparison theorems in differential equations - Theorem E and Theorem 2.2. This duality between two comparison theorems in differential equations of the same type, in turn gives rise to the duality in swapping differential equations of differ- ent types and hence generates Comparison Theorems on Differential Equations of Mixed Types I and II. Whereas, mixed type I concerns with comparing supersolu- tions of a Riccati type equation with subsolutions of a Jacobi type equation, mixed type II concerns with comparing supersolutions of a Jacobi type equation with subsolutions of a Riccati type equation.

2.4. Comparison Theorems on Differential Equations of Mixed Types. Theorem 2.3. (Comparing Differential Equations of Mixed Type I) Let functions Gi : (0,ti) (0, ) R satisfy (2.15) on (0,t1) (0,t2). Let g1 AC(0,t1) be a ⊂ ∞ → 1 ∩ ∈ solution of (2.24) and f C([0,t ]) C (0,t ) with f ′ AC(0,t ) be a positive 2 ∈ 2 ∩ 2 2 ∈ 2 solution of (2.14) with constants κi satisfying (2.16). Then t t and 1 ≤ 2 κ2f2′ (2.30) g1 ≤ f2 on (0,t1) .

Proof. Proceed as in the proof of Theorem 2.1, via (2.3) in which g = g2 ,f = f2 , and κ = κ2, the system (2.14) is transformed into (2.25) Applying Theorem 2.2 to (2.24) and (2.25), we have (2.26), and hence (2.30), via (2.3) on (0,t1) . 

Corollary 2.4. (1) Let Gi : (0,ti) (0, ) R, i = 1, 2 satisfy (2.15) on (0,t ) (0,t ), g AC(0,t ) be a solution⊂ of∞(2.27)→ , and f C([0,t ]) C1(0,t ) 1 ∩ 2 1 ∈ 1 2 ∈ 2 ∩ 2 with f ′ AC(0,t ) be a positive solution of (2.23). Then t t and 2 ∈ 2 1 ≤ 2 f2′ (2.31) g1 ≤ f2 on (0,t1) . (2) Theorem 2.3 is equivalent to (1). 18 S.W. WEI

Proof. (1) Choose κ = κ = 1 in Theorem 2.3. (2) Enough to show that (1) 1 2 ⇒ Theorem 2.3. Letg ˜ = g1 and f˜ = κ f . Theng ˜ satisfies (2.27) if and only if 1 κ1 2 2 2 1 g1 satisfies (2.24) and f˜2 satisfies (2.23) if and only if f2 satisfies (2.14). It follows ′ f˜2 from (1) thatg ˜1 ˜ . Hence, on (0,t1), ≤ f2 ˜ g1 f2′ f2′ =g ˜1 = κ1 ≤ f˜2 f2 ′ ′ f2 f2 Consequently, using (2.16) yields g1 κ1 κ2 , the desired (2.30) on (0,t1). f2 f2 ≤ ≤ 

Theorem 2.4. (Comparing Differential Equations of Mixed Type II) Let functions Gi : (0,ti) (0, ) R, i = 1, 2 satisfy (2.15) on (0,t1) (0,t2). Let f1 ⊂1 ∞ → ∩ ∈ C([0,ti]) C (0,t ) with f ′ AC(0,t ) be a positive solution of (2.13) and g ∩ 1 1 ∈ 1 2 ∈ AC(0,t2) be a solution of (2.25) with constants κi satisfying (2.16). Then t1 t2 and ≤

κ1f1′ (2.32) g2 f1 ≤ on (0,t1) .

Proof. Proceed as in the proof of Theorem 2.1, via (2.3) in which g = g1 ,f = f1 , and κ = κ1, the system (2.13) is transformed into (2.24). Applying Theorem 2.2 to (2.24) and (2.25), we have (2.26), and hence (2.32), via (2.3) on (0,t1) .  Similarly, we have

Corollary 2.5. (1) Let Gi : (0,ti) (0, ) R, 1 i 2 satisfy (2.15) on ⊂ 1 ∞ → ≤ ≤ (0,t1) (0,t2). Let f1 C([0,t2]) C (0,t1) with f1′ AC(0,t1) be a positive solution∩ of (2.22) and g∈ AC(0,t )∩ be a solution of (2.28)∈ . Then 2 ∈ 2 f1′ (2.33) g2 f1 ≤ on (0,t1) . (2) Theorem 2.4 is equivalent to (1).

2.5. Applications in Comparison Theorems in Riemannian Geometry. M When the radial Ricci curvature Ricrad of a manifold M has a lower bound, it can be viewed as a supersolution of Riccati type equation in disguise. Similarly, when the radial curvature K of a manifold has an upper bound (resp. a lower bound) , it can be viewed as a subsolution (resp. supersolution) of Riccati type equation in disguise. Thus, utilizing dualities in comparison theorems in differential equations of mixed types, we generate comparison theorems in Riemannian geometry and provide simple and direct proofs. Let M be an n-dimensional Riemannian manifold. M is said to be a manifold with a pole x0 , if D(x0)= M x0 . We recall the radial vector field ∂ on D(x0) is the unit vector field such that\{ for} any x D(x ), ∂ (x) is the unit vector tangent ∈ 0 to the unique geodesic γ joining x0 to x and pointing away from x0. A radial plane is a plane π which contains ∂(x) in the tangent space TxM. By the radial curvature K of a manifold, we mean the restriction of the sectional curvature function to all DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS19 the radial planes. We define K(t) to be the radial curvature of M at x such that r(x)= t. We also study Definition 2.1. The radial Ricci curvature of a manifold is the restriction of the M Ricci curvature function to all the radial vector fields. Denoted Ricrad(r), the radial Ricci curvature of M at x such that dist(x0, x)= r. Let a tensor g dr dr = 0 on the radial direction, and be just the metric − ⊗ tensor g on the orthogonal complement ∂⊥. At x M, the Hessian of r , de- ∈ noted by Hess r is a quadratic form on TxM given by Hess r (v, w) = ( vdr)w = ∇ g( v r, w) for v, w TxM. Here v is the covariant derivative in v direc- tion,∇ ∇ r is the gradient∈ vector field∇ of r , and hence is dual to the differential dr of ∇r . Thus, Hess r( r, r)=0 . The Laplacian of r , is defined to be ∆r = trace Hess r . We say∇ ∆r∇ f(r) holds weakly on M, if for every 0 ϕ(r) ≤ ≤ ∈ C∞(M) , ϕ(r)∆r dv ϕ(r)f(r) dv . We recall 0 M
≤ M Lemma A.R [24, Lemma 9.1]R (see [31], [41]) If ∆r f(r) holds pointwise in D(x0) , where f C0(0, ) , then ∆r f(r) holds weakly≤ on M. ∈ ∞ ≤ 2.6. Under radial Ricci curvature assumptions.

Theorem 2.5. (Laplacian Comparison Theorem) Let functions Gi : (0,ti) (0, ) R, i =1, 2 satisfy (2.15) on (0,t ) (0,t ). Assume ⊂ ∞ → 1 ∩ 2 (2.34) (n 1) G (r) RicM (r) − 1 ≤ rad ◦ 1 on Bt1 (x0) D(x0) , and let f2 C([0,t2]) C (0,t2) with f2′ AC(0,t2) be a positive solution⊂ of ∈ ∩ ∈

f ′′ + G f 0 a. e. in (0,t ) (2.35) 2 2 2 ≥ 2 f (0) = 0 ,f ′ (0) = n 1 . ( 2 2 − Then t t and 1 ≤ 2 f (2.36) ∆r (n 1) 2′ (r) ≤ − f2

◦ in Bt1 (x0) . If in addition, (2.34) occurs in D(x0) , then (2.27) holds pointwise on D(x0) and weakly on M. Proof. For a bilinear form , we write 2 = trace( t) . Recall A |A| AA Theorem F. (Weitzenb¨ock formula) For every function f C3(M) , ∈ 1 (2.37) ∆ f 2 = Hess f 2 + ∆f, f + Ric f, f 2 |∇ | | | h∇ ∇ i ∇ ∇ Substituting the distance function r(x) for f(x) into Weitzenb¨ock
formula (2.37), we have inside the cut locus of x ( r = 1), via Gauss lemma 0 |∇ | ∂ ∂ ∂ 0= Hess r 2 + ∆r + Ric( , ) | | ∂r ∂r ∂r 2 Since Hess( ∂ , ∂ )=0 , Hess r 2 (∆r) , Ric( ∂ , ∂ ) = RicM (r) and (2.34) ∂r ∂r | | ≥ n 1 ∂r ∂r rad occurs, it follows that −

2 ∂ ∆r + (∆r) + (n 1)G (r) 0 on B◦ (x ) (2.38) ∂r n 1 1 t1 0 n 1 − − +≤ (∆r = − + O(1) as t 0 .
r → 20 S.W. WEI

Let g = ∆r, κ = n 1 = κ . Then (2.38) becomes (2.24) and (2.35) becomes 1 1 − 2 (2.14). Via (2.24) and (2.35) with Gi satisfying (2.15) on (0,t ) (0,t ) and κ 1 ∩ 2 1 satisfying (2.16). Applying Theorem 2.3 we have (2.30) holds on (0,t1), i.e., (2.36) ◦ holds in Bt1 (x0) . If in addition, (2.34) occurs in D(x0) , then (2.36) holds pointwise on D(x0). By Lemma A or using Green’s Identity and a double limiting argument (see [24, Lemma 9.1],[31], [41]), (2.36) holds weakly on M. 

In applying Theorem 2.5, when a specific Ricci curvature lower bound assump- tion is given, we do not need to assume f2 in (2.35) for comparison. Instead, we ′ f2 find f2 in (2.44) and estimate in (2.49) and obtain: f2 Theorem 2.6. (1) If A(A 1) (2.39) (n 1) − RicM (r), where A 1 − − r2 ≤ rad ≥

◦ on Bt (x ) D(x ), then 1 0 ⊂ 0 A (2.40) ∆r (n 1) ≤ − r

◦ on Bt1 (x0). If in addition, (2.39) occurs on D(x0) , then (2.40) holds pointwise on D(x0) and weakly on M. (2) Equivalently, if A(A 1) (2.41) (n 1) − RicM (r), A 1 − − (c + r)2 ≤ rad ≥

◦ ◦ on Bt (x ) D(x ), where c 0 , then (2.40) holds on Bt (x ). If in addition 1 0 ⊂ 0 ≥ 1 0 (2.41) occurs on D(x0), then (2.40) holds pointwise on D(x0) and weakly on M. Proof. (2) (1) Apparently (1) is a special case c = 0 in (2). (1) (2) If (2.41) ⇒ A(A 1) ⇒ A(A 1) occurs, then (2.39) takes place, since [ (n 1) − , ) [ (n 1) − , ) . − − (c+r)2 ∞ ⊂ − − r2 ∞ We then apply (1) . Now we prove (2). First assume (2.41) occurs for c> 0. Choose A A 1 φ = (n 1)r , where A 1 . Then φ′ = (n 1)Ar − . Hence, for r> 0 − ≥ − φ A (2.42) ′ = φ r A 2 and φ′′ = (n 1)A(A 1)r − . That is, for r> 0 , φ is a solution of the following differential equation:− − A(A 1) (2.43) φ′′ + − − φ =0 . r2

Let f2 be a positive solution of f + G f =0 a. e. in (0,t ) (2.44) 2′′ 2 2 2 (f2(0) = 0 ,f2′ (0) = κ2 ,

A(A 1) where G = − − , κ = n 1, and 2 (c+r)2 2 − (2.45) t = sup r : f > 0 on (0, r) . 2 { 2 } DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS21

We note t = . This can be seen by comparing the solution f of (2.44) with the 2 ∞ 2 solution f˜ (r) = (n 1)r of the following 2 − f˜ ′′ +0 f˜ =0, 2 · 2 ˜ ˜ (f2(0) = 0 , f2′(0) = n 1 , − and applying a standard Sturm comparison theorem. Furthermore,

(2.46) (φ′f f ′ φ)(0) = 0 . 2 − 2 A(A 1) A(A 1) In view of (2.43), (2.44), (2.45), and f φ − − 0, for r (0, ) 2 r2 − (c+r)2 ≥ ∈ ∞ (φ′f2 f ′ φ)′ = φ′′f2 f ′′φ
(2.47) − 2 − 2 0 ≥ The monotonicity then implies that

(2.48) φ′f f ′ φ 2 ≥ 2 for r (0, ) which in turn via (2.42) yields ∈ ∞ f φ A (2.49) 2′ ′ = f2 ≤ φ r A(A 1) A(A 1) on (0, ) . Applying comparison Theorem 2.5 in which G1(r)= −(c+r−)2 −(c+r−)2 = ∞ ≥ ′ f2 G2 (c > 0) , where G1 is defined in (0,t1) , G2 is defined on (0, ) and is esti- ∞ f2 ◦ mated as in (2.49), we have shown (2.40) holds on Bt1 (x0) under (2.41) for every c > 0 . Now we prove (1) by passing c 0 in (2.41) in the following way: For → ◦ ◦ every x Bt (x ), there exist sequences xi in a radial geodesic ray in Bt (x ) ∈ 1 0 { } 1 0 and ci > 0 such that xi x,ci 0 , and r(xi)+ ci = r(x) . It follows that for { } → → every ci > 0 ,

M A(A 1) A Ricrad(r)(xi) (n 1) − 2 ∆r(xi) (n 1) (xi) , ≥− − r(xi)+ ci ⇒ ≤ − r (2.50) and hence , as i , → ∞
M A(A 1) A Ricrad(r)(x) (n 1) − 2 ∆r(x) (n 1) (x) . ≥− − r(x) ⇒ ≤ − r

◦
This proves that (2.40) holds on Bt (x ) under (2.41) for c 0 , or under (2.39) 1 0 ≥ ◦ on Bt1 (x0). If in addition (2.39) or (2.41) occurs on D(x0) , then applying Lemma A (Lemma 9.1 in [24]), or a double limiting argument (see [41, Sect. 3]), we obtain the desired (2.40) pointwise on D(x0) , and weakly on M. This proves (1) (2), and (1) . ⇒  Theorem 2.7. (1) If B (1 B ) (2.51) (n 1) 1 − 1 RicM (r), where 0 B 1 − r2 ≤ rad ≤ 1 ≤ ◦ on Bt (x ) D(x ), then 1 0 ⊂ 0 1+ 1+4B (1 B ) (2.52) ∆r (n 1) 1 − 1 ≤ − p 2r 22 S.W. WEI

◦ in Bt1 (x0). If in addition, (2.51) occurs on D(x0) , then (2.52) holds pointwise on D(x0) and weakly on M. (2) Equivalently, if B (1 B ) (2.53) (n 1) 1 − 1 RicM (r) , 0 B 1 − (c + r)2 ≤ rad ≤ 1 ≤

◦ ◦ on Bt (x ) D(x ) , where c 0 , then (2.52) holds in Bt (x ). If in addition 1 0 ⊂ 0 ≥ 1 0 (2.51) occurs on D(x0), then (2.52) holds pointwise on D(x0) and weakly on M. Proof. (2) (1) Obviously, (1) is a special case c = 0 in (2). To prove (1) (2), ⇒ α ⇒ enough to show (2.52) holds under (2.53), for any c > 0. Choose φ1 = r , where 1+√1+4B1(1 B1) − α 1 2 α = 2 . Then φ1′ = αr − and (2α 1) =1+4B1(1 B1) , i.e. α(α 1) = B (1 B ) . Hence, − − − 1 − 1 φ α 1+ 1+4B (1 B ) (2.54) 1′ = = 1 − 1 φ1 r p 2r α 2 φ′′ = α(α 1)r − , i.e., for r> 0 , 1 − B1(1 B1) (2.55) φ′′ − φ =0 . 1 − r2 1 B (1 B ) Let f > 0 satisfy (2.44), where G = 1 − 1 , κ = n 1 and t is as in 2 2 − (c+r)2 2 − 2 (2.45). Then t2 = , by applying the standard Sturm comparison theorem as in the proof of Theorem∞ 2.6. Similarly, we have (2.46). (2.47) and (2.48) hold for B1(1 B1) r (0, ) in which φ = φ1, since f2φ1 r−2 + G2 0 , for r (0, ) . It follows∈ from∞ (2.54) that ≥ ∈ ∞
f φ 1+ 1+4B (1 B ) (2.56) 2′ 1′ = 1 − 1 f2 ≤ φ1 p 2r on (0, ) . Applying comparison Theorem 2.5, in which G1 G1 : (0,t1) R = B (1 B∞) B (1 B ) → 1 − 1 1 − 1 = G G : (0,t ) R ,c > 0 on (0,t ) (0,t ), α = (c+r)2 ≥ − (c+r)2 2 2 2 → 1 ∩ 2
B B ′ B B 1+√1+4 1(1 1) f2 1+√1+4 1(1 1) ◦ − and − , we have
shown (2.52) holds on Bt (x0) 2 f2 ≤ 2r 1 under (2.53) for any c > 0. Proceeding the limiting argument as in the proof of Theorem 2.6 and (2.50), we prove similarly (1) (2), and (1) .  ⇒ Corollary 2.6. If the radial curvature K satisfies (2.57) A(A 1) A(A 1) − K(r) or equivalently, − K(r) , c 0 where 1 A − r2 ≤ − (c + r)2 ≤ ≥ ≤
◦ ◦ on Bt (x ) D(x ), then (2.40) holds on Bt (x ), and if in addition (2.57) occurs 1 0 ⊂ 0 1 0 on D(x0) , then (2.43) holds pointwise on D(x0) and weakly on M. Corollary 2.7. If the radial curvature K satisfies (2.58) B (1 B ) B (1 B ) 1 − 1 K(r) or equivalently, 1 − 1 K(r) , c 0 where 0 B 1 r2 ≤ (c + r)2 ≤ ≥ ≤ 1 ≤
◦ ◦ on Bt (x ) D(x ) , then (2.52) holds on Bt (x ), and if in addition (2.58) 1 0 ⊂ 0 1 0 occurs on D(x0) , then (2.52) holds pointwise on D(x0) and weakly on M. DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS23

2.7. Under the radial curvature assumptions. The following Theorem strength- ens main theorems in [24, Theorem 4.1], and [38, Theorem D]. We also give direct and simple proofs with applications from the view point of dualities.. Theorem 2.8. (Hessian and Laplacian Comparison Theorems) Let (2.59) G (r) K(r) 1 ≤ ◦ on Bt (x ) D(x ) resp. 1 0 ⊂ 0 (2.60) K(r) G (r) ≤ 2 ◦ 1 on Bt˜ (x0) D(x0) , and let f2 C([0,tf2]) C (0,t2) with f2′ AC(0,t2) be a 2 ⊂ ∈ ∩ ∈ positive solution of (2.14), where Gi : (0,ti) R satisfy (2.15) on (0,t1) (0,t2),
˜ 1 →˜ ˜ ∩ where 1 κ2 . resp. let f1 C([0, t1]) C (0, t1) with f1′ AC(0, t1) be a positive solution≤ of ∈ ∩ ∈ f ′′ + G f 0 on(0, t˜ ) (2.61) 1 1 1 ≤ 1 (f1(0) = 0 ,f1′ (0) = κ1 , f where Gi : (0, t˜i) R satisfy → (2.62) G G f 2 ≤ 1 on (0, t˜ ) (0, t˜ ), where 0 <κ 1 . 1 2 1 f f Then t ∩ t , ≤ 1 ≤ 2
κ f κ f (2.63) Hess r 2 2′ (g dr dr) and ∆r (n 1) 2 2′ (r) ≤ f2 − ⊗ ≤ − f2

◦ on Bt (x ) resp. t˜ t˜ , 1 0 1 ≤ 2 κ1f ′ κ1f ′ (2.64) 1 g dr dr Hess r and (n 1) 1 (r) ∆r f1 − ⊗ ≤ − f1 ≤
◦ on Bt˜1 (x0) in the sense of quadratic forms. If in addition (2.59) occurs on D(x0), then the second part of (2.63) holds pointwise on D(x ) and weakly in M.
0 Proof. Following the notation in [24]. Let γ be the unit speed geodesic curve joining x0 = γ(0) to x = γ(t0) , and V be a parallel vector field along γ(t) , for 0 t t0 . A direct computation shows that ≤ ≤ d (2.65) V r, V + V r, V r = R(V, r) r, V G dth∇ ∇ i h∇ ∇ ∇ ∇ i −h ∇ ∇ i≤− 1 (see [24, p.174, (4.5)]). Define Hess r(v, v) λmax(x)= max . v Tx(M) 0 ,v r(x) v, v { ∈ \{ } ⊥∇ } h i Select a unit vector v at x = γ(t0) such that

v r, v := Hess r(v, v)= λ γ(t ) . h∇ ∇ i max ◦ 0 Then 2 v r, v r = λ γ(t ) . h∇ ∇ ∇ ∇ i max ◦ 0 24 S.W. WEI

Let g = λ γ , and let the parallel vector field V along γ satisfying V (t )= v . 1 max ◦ 0 Then the function Hess r(V, V ) g1(t) 0 , attains its maximum value 0 at t = t0 , and at this point − ≤ d d Hess r(V, V )= g (t) . dt dt 1 t=t0 t=t0

It follows from (2.65) that g1 satisfies (2.27) (see [24, p.175]). By assumption, f2 is a positive solution of (2.14), with κ2 1 , and (2.15) holds on (0,t1). Applying Theorem 2.3 in which g = λ γ, we≥ have 1 max ◦ κ2f2′ Hess r(w, w) Hess r(v, v)= g1 ≤ ≤ f2

◦ on Bt (x ) for any unit vector w r(x) at x . Taking the trace, we obtain the 1 0 ⊥∇ ◦ desired (2.63) on Bt1 (x0). If in addition (2.59) occurs on D(x0), then the second part of (2.63) holds weakly in M by Lemma A. Similarly, let V˜ be a parallel vector field along γ(t) , for 0 t t0 . By the radial curvature assumption (2.60), ≤ ≤ d (2.66) ˜ r, V˜ + ˜ r, ˜ r = R(V,˜ r) r, V˜ G . dth∇V ∇ i h∇V ∇ ∇V ∇ i −h ∇ ∇ i≥− 2 Define f Hess r(v, v) λmin(x) = min . v Tx(M) 0 ,v r(x) v, v { ∈ \{ } ⊥∇ } h i Select a unit vectorv ˜ at x = γ(t0) such that

v r, v˜ := Hess r(˜v, v˜)= λ γ(t ) . h∇˜∇ i min ◦ 0 Then 2 v r, v r = λ γ(t ) . h∇˜∇ ∇˜∇ i min ◦ 0 Let g = λ γ , and let the parallel vector field V˜ along γ satisfying V˜ (t )=˜v . 2 min ◦ 0 Then the function Hess r(V,˜ V˜ ) g2(t) 0 , attains its minimum value 0 at t = t0 , and at this point − ≥ d d Hess r(V,˜ V˜ )= g (t) . dt dt 2 t=t0 t=t0

It follows from (2.66) that g2 satisfies

2 g2 ′ + g2 + G2 0 a. e. in (0, t˜2) (2.67) 1 ≥ + (g2(t)= t + O(1) as t 0 , f → We note f is a positive solution of (2.61), where 0 <κ 1. (see [24, p.176]) and 1 1 ≤ (2.62) holds on (0, t˜ ) (0, t˜ ). Applying Theorem 2.4, in which G = G˜ ,t = t˜ , 1 ∩ 2 1 1 1 1 G2 = G˜2,t2 = t˜2 and κ2 = 1, we have

κ1f1′ g2 = Hess r(˜v, v˜) Hess r(w, w) f˜1 ≤ ≤

◦ on B ˜ (x0) for any unit vector w r(x) at x . Taking the trace, we obtain the t1 ⊥∇ ◦  desired (2.64) on Bt˜1 (x0). DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS25

˜ 1 ˜ ˜ Remark 2.1. If f1 C([0, t1]) C (0, t1) with f1′ AC(0, t1) is a positive solution of ∈ ∩ ∈ f + G f =0 on (0, t˜ ) (2.68) 1′′ 1 1 1 (f1(0) = 0 ,f1′ (0) = κ1 f then obviously f1 is a positive solution of (2.61) Hence, we are ready for applying Theorem 2.8. The advantage of (2.68) over (2.61) is that a solution f1 of (2.68) will make the subsequent (3.11), the monotonicity of φ′f1 f1′ φ, and the estimate (3.41) work. −

Corollary 2.8. (1) Let the radial curvature K satisfy (2.59) resp. (2.60) , and

1  1 let f C([0,t ]) C (0,t ) with f ′ AC(0,t ) resp. f C([0, t˜ ]) C (0, t˜ ) 2 ∈ 2 ∩ 2 2 ∈ 2 1 ∈ 1 ∩ 1 with f ′ AC(0, t˜ ) be a positive solution of (2.23) resp. (2.61),κ = 1 . As- 1 ∈ 1 1  
sume (2.15) occurs on (0,t ) (0,t ) resp. (2.62) occurs on (0, t˜ ) (0, t˜ ) . Then 1 ∩ 2 1 ∩ 2   ◦ ◦ (2.63) holds on Bt1 (x0) for κ2 =1 resp. (2.64) holds on Bt˜1 (x0) for κ1 =1 . If   in addition (2.59) occurs on D(x0), then the second part of (2.63) (2) Theorem 2.8 is equivalent to (1) .

2.8. Under the sectional curvature and Ricci curvature assumptions. Corollary 2.9. Denote RicM the Ricci curvature of M and SecM the sectional M M curvature of M. If we replace “ Ricrad ” in Theorems 2.5, 2.6 and 2.7 by “ Ric ” , the results remain to be true. Likewise, if we replace “K or K(r)” in Theorems 2.8, 3.1, 3.2, 3.3, 3.4, 3.5, and Corollaries 2.6 - 2.8, 3.1 - 3.5 by “SecM ” , the results remain to be true. Proof. This follows at once from Definition 2.1, the definition of the radial curva- ture, and the above Theorems and Corollaries stated in Corollary 2.9. 

3. Curvature in Comparison Theorems Hessian and Laplacian Comparison Theorem 2.8 has many geometric applica- tions under curvature assumptions. Let A , A1 ,B,B1 be constants and K(r) be the radial curvature of M. Theorem 3.1. (1) If A(A 1) (3.1) − K(r) where A 1 − r2 ≤ ≥

◦ on Bt (x ) D(x ), then 1 0 ⊂ 0 A (3.2) Hess r g dr dr in the sense of quadratic forms ≤ r − ⊗   and A (3.3) ∆r (n 1) ≤ − r 26 S.W. WEI

◦ hold on Bt1 (x0). If in addition (3.1) occurs on D(x0) , then (3.2) and (3.3) hold pointwise on D(x0) , and (3.3) holds weakly on M. (2) Equivalently, if A(A 1) (3.4) − K(r) , A 1 − (c + r)2 ≤ ≥

◦ ◦ on Bt (x ) D(x ), where c 0 , then (3.2) and (3.3) hold on Bt (x ). If in 1 0 ⊂ 0 ≥ 1 0 addition (3.4) occurs on D(x0), then (3.2) and (3.3) hold pointwise on D(x0) , and weakly on M. Proof of Theorem 3.1. (2) (1) (1) is the special case c = 0 in (2). (1) (2) ⇒ A(A 1) A(A 1) ⇒ If (3.4) occurs, then (3.1) takes place, since [ − , ) [ − , ) . Then − (c+r)2 ∞ ⊂ − r2 ∞ apply (1) . Now we prove (2). First assume (3.4) occurs for c > 0. Proceed as in A A 1 the proof of Theorem 2.6, choose φ = r , where A 1 . Then φ′ = Ar − . Hence, ≥ for r> 0 , we have (2.42) and (2.43). Let f2 be a positive solution of (2.44), where A(A 1) G = − − ,κ = 1 and t be as in (2.45). We note t = , by a standard Sturm 2 (c+r)2 2 2 2 ∞ comparison theorem. Furthermore, (2.46) (φ′f2 f2′ φ)(0) = 0 holds. In view of − A(A 1) A(A 1) (2.43), (2.44) and (2.45), we have (2.47) (φ′f f ′ φ)′ = f φ 2− − 2 0 , 2 − 2 2 r − (c+r) ≥ for r (0, ) . The monotonicity then implies that φ f f φ for r (0, ) which ′ 2 2′
in turn∈ via∞ (2.42) yields ≥ ∈ ∞ f φ A (2.49) 2′ ′ = f2 ≤ φ r A(A 1) A(A 1) on(0, ) . Applying comparison Theorem 2.8 in which G = − − − − = ∞ 1 (c+r)2 ≥ (c+r)2 ′ f2 A ◦ G2 (c> 0) ,κ2 =1 , , and taking the trace, we have shown on Bt (x0) , (2.63) f2 ≤ r 1 holds, i.e., (3.2) and (3.3) hold under (3.4) for every c> 0. Proceeding the limiting argument as in (2.50) and applying Lemma A, we prove similarly (1) (2) , and (1) . ⇒  Theorem 3.2. (1) If A (A 1) (3.5) K(r) 1 1 − on M x where A 1 , ≤− r2 \{ 0} 1 ≥ then Hess r and ∆r satisfy A A (3.6) 1 g dr dr Hess r and (n 1) 1 ∆r r − ⊗ ≤ − r ≤   on M x0 , respectively. (2) Equivalently,\{ } if A (A 1) (3.7) K(r) 1 1 − , A 1 ≤− (c + r)2 1 ≥ on M x where c 0 , then (3.6) holds. \{ 0} ≥ Proof of Theorem 3.2. (2) (1) (1) is the special case c = 0 in (2). (1) (2) ⇒ ⇒ Enough to show (2) holds for the case c> 0 in (3.7). Choose φ = (c + r)A1 , where A 1 A 1 . Then φ′ = A (c + r) 1 − . Hence, for r> 0 1 ≥ 1 φ A (3.8) ′ = 1 φ c + r DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS27

A 2 and φ′′ = A (A 1)(c + r) 1− . That is, for r> 0 , φ is a solution of the equation: 1 1 − A1(A1 1) (3.9) φ′′ + − − φ =0 . (c + r)2

Let f1 be a positive solution of f + G f =0 on (0, t˜ ) (2.68) 1′′ 1 1 1 (f1(0) = 0 ,f1′ (0) = κ1 f A1(A1 1) where G1 = − (c+r)−2 ,κ1 = 1 and

(3.10) f t˜ = sup r : f > 0 on (0, r) 1 { 1 } We note t˜1 = , by the same standard Sturm comparison theorem. In view of (3.9), (2.68) and∞ (3.10), for r (0, ) ∈ ∞

A1(A1 1) A1(A1 1) (φ′f1 f ′ φ)′ = f1φ − − (3.11) − 1 (c + r)2 − (c + r)2 =0 .

A1 Since (φ′f1 f1′ φ)(0) = c 0 , the monotonicity then implies that φ′f1 f1′ φ on (0, ) which− in turn via− (3.8)≤ yields ≤ ∞ A φ f (3.12) 1 = ′ 1′ c + r φ ≤ f1

A1(A1 1) on(0, ) . Applying Remark 2.1 and Theorem 2.8 in which G1(r) = − (c+r)−2 = ∞ ′ f1 A1 G2(r)(c> 0),κ1 =1, and t˜1 = we have shown via (2.64) that f1 ≥ c+r ∞ f A A (3.6)f 1 g dr dr Hess r and (n 1) 1 ∆r c + r − ⊗ ≤ − c + r ≤   on M x , under (3.7) for every c> 0. Now proceed analogously to the proof of \{ 0} Theorem 2.6 : For every x M x , there exist sequences xi in a radial geodesic ∈ \{ 0} { } ray in M (x ) and ci > 0 such that xi x,ci 0 , and r(xi)+ ci = r(x) . It \ 0 { } → → follows that for every ci > 0 ,

A1(A1 1) A1 K(r)(xi) − 2 ∆r(xi) (n 1) , ≤− ci + r(xi) ⇒ ≥ − ci + r(xi) (3.13) and hence , as i ,
→ ∞ A1(A1 1) A1 K(r)(x) −2 ∆r(x) (n 1) (x) , ≤− r(x) ⇒ ≥ − r

A
and similarly, Hess r(x) 1 g dr dr (x) for every x M x . This proves ≥ r − ⊗ ∈ \{ 0} (1) (2) and (1).   ⇒ 

Corollary 3.1. (1) If the radial curvature K satisfies A(A 1) A (A 1) (3.14) − K(r) 1 1 − where A A 1 − r2 ≤ ≤− r2 ≥ 1 ≥ 28 S.W. WEI on M x0 , then (3.15)\{ } A A 1 g dr dr Hess r g dr dr in the sense of quadratic forms, r − ⊗ ≤ ≤ r − ⊗     A A (n 1) 1 ∆r (n 1) pointwise on M x and − r ≤ ≤ − r \{ 0} A ∆r (n 1) weakly on M. ≤ − r (2) Equivalently, if K satisfies A(A 1) A (A 1) (3.16) − K(r) 1 1 − , A A 1 − (c + r)2 ≤ ≤− (c + r)2 ≥ 1 ≥ on M x , where c 0 , then (3.15) holds. \{ 0} ≥ Proof. This follows at once from Theorems 3.1 and 3.2. 

Theorem A. (1)([24]) Let the radial curvature K satisfy A A (3.17) K(r) 1 where 0 A A − r2 ≤ ≤− r2 ≤ 1 ≤ on M x0 . Then (3.18)\{ } 1+ √1+4A 1+ √1+4A 1 g dr dr Hess(r) g dr dr on M x , 2r − ⊗ ≤ ≤ 2r − ⊗ \{ 0}     (3.19) 1+ √1+4A 1+ √1+4A (n 1) 1 ∆r (n 1) pointwise on M x , and − 2r ≤ ≤ − 2r \{ 0}

1+ √1+4A (3.20) ∆r (n 1) weakly on M. ≤ − 2r (2) Equivalently, if A A (3.21) K(r) 1 where 0 A A − (c + r)2 ≤ ≤−(c + r)2 ≤ 1 ≤ on M x , then (3.18)-(3.20) hold. \{ 0} 2 2 Proof of Theorem A. Let A(A 1) = a in (3.1) resp. A1(A1 1) = a1 in (3.5) . − 2 − 1+√1+4a2 1+√1+4a1 A 1+√1+4a2 Then A = (resp. A1 = ) 1. Hence, = (resp.
2 2 ≥ r 2r a2 A1 1+√1+4 1 r = 2r ). Substitute these into Theorem 3.1 (resp. Theorem 3.2) so that 2 2 2 2 this Theorem is rephrased in terms of a (resp. a1). Replacing a (resp. a1) in this rephrase by A (resp. A1), we transform Corollary 3.1 into Theorem A. 

Corollary 3.2. (1) If the radial curvature K satisfies A(A 1) A (3.22) K(r)= − , where 1 A resp.K(r)= , where 0 A − r2 ≤ −r2 ≤
DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS29 on M x , then \{ 0} A 1+ √1+4A Hess r = resp. Hess r = (g dr dr) and (3.23) r 2r − ⊗ A 1+ √1+4A
∆r = (n 1) resp. ∆r = (n 1) − r − 2r on M x0 .
(2) Equivalently,\{ } if K satisfies A(A 1) A (3.24) K(r)= − , where 1 A resp.K(r)= , where 0 A − (c + r)2 ≤ −(c + r)2 ≤ on M x , and c 0 , then (3.23) holds on M x .
\{ 0} ≥ \{ 0} Proof. This follows at once from combining Theorems 3.1 and 3.2 in which A1 = A.  Theorem 3.3. (1) If B (1 B ) (3.25) 1 − 1 K(r) where 0 B 1 r2 ≤ ≤ 1 ≤ ◦ on Bt (x ) D(x ), then Hess r and ∆r satisfy 1 0 ⊂ 0 1+ 1+4B (1 B ) (3.26) Hess r 1 − 1 g dr dr and ≤ 2r − ⊗ p   1+ 1+4B (1 B ) (3.27) ∆r (n 1) 1 − 1 ≤ − p 2r ◦ on Bt (x ) D(x ) respectively. If in addition (3.25) occurs on D(x ) , then (3.26) 1 0 ⊂ 0 0 holds pointwise on D(x0) and (3.27) holds weakly on M. (2) Equivalently, if B (1 B ) (3.28) 1 − 1 K(r), 0 B 1 (c + r)2 ≤ ≤ 1 ≤

◦ ◦ on Bt (x ) D(x ), where c 0 , then (3.26) and (3.27) hold on Bt (x ). If in 1 0 ⊂ 0 ≥ 1 0 addition (3.28) occurs on D(x0), then (3.26) holds pointwise on D(x0) and (3.27) holds weakly on M. Proof. (2) (1) (1) is the special case c = 0 in (2). To prove (1) (2), enough to ⇒ ⇒ ◦ ◦ show that on Bt1 (x0) , (3.26) and (3.27) hold under (3.28), on Bt1 (x0) for any c> 0, since we can combine this with (1) . Proceed as in the proof of Theorem 2.7, choose

1+√1+4B1(1 B1) α − α 1 φ1 = r , where α = 2 , 0 B1 1 . Then α 1 , φ1′ = αr − and 2 ≤ ≤ ≥ (2α 1) =1+4B1(1 B1) , i.e. α(α 1) = B1(1 B1) . Hence, for r> 0 , we have ′ φ1 − α − α 2 − − = , and φ′′ = α(α 1)r − , i.e. (2.54) and (2.55) hold. Let f2 be a positive φ1 r 1 − B (1 B ) solution of (2.44), in which G = 1 − 1 ,κ = 1 and t be as in (2.45). Then 2 − (c+r)2 2 2 by the standard comparison theorem as before, t = . 2 ∞ ◦ Furthermore, we have (2.46), (2.47) and (2.48) hold on Bt1 (x0) in which φ = φ1, B1(1 B1) since f2φ1 r−2 + G2 0 , for r (0, ) . It follows from (2.54) that (2.56), ′ ′ ≥ ∈ ∞ f φ 1+√1+4B1(1 B1) i.e., 2 1 =
− holds on (0, ) . f2 ≤ φ1 2r ∞ 30 S.W. WEI

B (1 B ) B (1 B ) Applying comparison Theorem 2.8, in which G = 1 − 1 1 − 1 = 1 (c+r)2 ≥ − (c+r)2 ′ B B f2 1+√1+4 1(1 1) ◦ G2 (c> 0) ,κ2 =1 , − , we have shown that on Bt (x0) , (2.63) f2 ≤ 2r 1 holds, i.e., (3.26) and (3.27) hold under (3.28) for any c> 0. This proves (1) (2). ⇒ ◦ This also proves (1), since (3.25) occurs on Bt1 (x0) implies that (3.28) happens on ◦ Bt1 (x0) for any c > 0. If in addition (3.25) occurs on D(x0), then (3.26) holds pointwise on D(x0) and (3.27) holds weakly on M by Lemma A ([24, Lemma 9.1]), or a double limiting argument (see [41, Sect. 3]).  α Remark 3.1. In proving Theorem 3.3, we choose axillary function φ1 = r , where 1+√1+4B1(1 B1) − α = 2 . We may choose, for example a different axillary function 1+√1 4B1(1 B1) α1+1 − − φ1 = r , where α1 = 2 , and obtain estimates 1 3 B1 + Hess r | − 2 | 2 g dr dr and ≤ r − ⊗ (3.29)   1 3 B1 2 + 2 ◦ ∆r (n 1)| − | pointwise on Bt (x ) D(x ) ≤ − r 1 0 ⊂ 0 If in addition (3.25) occurs on D(x0) , then 1 3 B1 + (3.30) ∆r (n 1)| − 2 | 2 weakly on M. ≤ − r

1+√1+4B1(1 B1) − However, the upper bound estimate 2r obtained in (3.26) and (3.27) B 1 + 3 α | 1− 2 | 2 by selecting φ1 = r is better than the upper bound estimate r obtained 1+√1+4B1(1 B1) α1 +1 − B1+1 in (3.29) and (3.30) by selecting φ1 = r . Indeed, 2r r = B 1 + 3 B 1 + 3 ≤ 1− 2 2 | 1− 2 | 2 r r . For the completeness or comparison, we include the following derivation≤ for the estimates (3.29) and (3.30). α1 +1 α1 Note (2α1 1) = 2B1 1 . Analogously, φ1 = r implies φ1′ = (α1 + 1)r . For r> 0 , − | − | 1 3 φ α +1 B1 + (3.31) 1′ = 1 = | − 2 | 2 φ1 r r

B1 1 and φ1′′ = (α1 + 1)α1r − , i.e.

(α1 + 1)α1 (3.32) φ′′ + − φ =0 . 1 r2 1

(α1+1)α1 Let f2 be a positive solution of (2.44), where G2 = − (c+r)2 ,κ2 =1 and let t2 be as in (2.45). Then t = , (φ′ f f ′ φ )(0) = 0 , and for r (0, ) 2 ∞ 1 2 − 2 1 ∈ ∞ α1(α1 + 1) (φ′ f f ′ φ )′ = f φ + G 0 . 1 2 − 2 1 2 1 r2 2 ≥ Monotonicity implies
1 3 f φ B1 + 2′ 1′ = | − 2 | 2 on (0, ) f2 ≤ φ1 r ∞ B (1 B ) (α +1)α Applying Theorem 2.8, in which G = 1 − 1 0 − 1 1 = G ,κ = 1 (c+r)2 ≥ ≥ (c+r)2 2 2 f ′ B 1 + 3 1 , 2 = | 1− 2 | 2 , and taking the trace, we obtain the estimates (3.29). Applying f2 r Lemma A, if in addition, (3.25) occurs on D(x0), (3.30) follows . DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS31

Corollary 3.3. If the radial curvature K satisfies B B 1 (3.33) 1 K(r) resp. 1 K(r) ,c 0 , where 0 B r2 ≤ (c + r)2 ≤ ≥ ≤ 1 ≤ 4
◦ on Bt (x ) D(x ) , then 1 0 ⊂ 0 1+ √1+4B 1+ √1+4B (3.34) Hess r 1 g dr dr and ∆r (n 1) 1 ≤ 2r − ⊗ ≤ − 2r   ◦ holds pointwise on Bt1 (x0) . If in addition, (3.33) holds in D(x0), then the second part of (3.34) holds weakly on M.

Proof. This follows immediately from substituting B1(1 B1) in Theorem 3.3 for B , and B (1 B )= (B 1 )2 + 1 1 , B (1 B ) − 0 if 0 B 1 .  1 1 − 1 − 1 − 2 4 ≤ 4 1 − 1 ≥ ≤ 1 ≤ Theorem 3.4. (1) If B(1 B) (3.35) K(r) − , 0 B 1 ≤ r2 ≤ ≤

◦ on Bt (x ) D(x ), then Hess r and ∆r satisfy 2 0 ⊂ 0 B 1 + 1 | − 2 | 2 g dr dr Hess r and r − ⊗ ≤ (3.36)   B 1 + 1 (n 1)| − 2 | 2 ∆r − r ≤

◦ on Bt2 (x0) respectively. (2) Equivalently, if B(1 B) (3.37) K(r) − , 0 B 1 ≤ (c + r)2 ≤ ≤

◦ ◦ on Bt (x ) D(x ) , where c 0 , then (3.36) holds on Bt (x ) . 2 0 ⊂ 0 ≥ 2 0 Proof. (2) (1) (1) is the special case c = 0 in (2). (1) (2) If (3.37) occurs, ⇒ B(1 B) B(1 B) ⇒ then (3.35) occurs, since ( , − ] ( , − ] . We then apply (1) . To −∞ (c+r)2 ⊂ −∞ r2 1+√1 4B(1 B) − − prove (2), we first assume (3.37) occurs for c > 0. Choose β = 2 , with 0 B 1 , and φ = rβ . Then 2β 1= (2B 1)2 , i.e. ≤ ≤ 2 − − 1 1 p B if 1 B (3.38) β = B + = max B, 1 B = 2 ≤ | − 2| 2 { − } 1 B if 0 B < 1 , ( − ≤ 2 β 1 β(β 1) = B(B 1) and φ′ = βr − for r> 0 . Hence, − − 2 φ β (3.39) 2′ = φ2 r β 2 for r> 0 , and for 0 B 1 , φ′′ = B(1 B)r − , i.e. ≤ ≤ 2 − − B(1 B) (3.40) φ′′ + − φ =0 . 2 r2 2 32 S.W. WEI

B(1 B) ˜ Let f1 be a positive solution of (2.68), where G1 = (c+−r)2 ,κ1 = 1 and let t1 be as in (3.10). Then f (φ′ f f ′ φ )(0) = 0 if β =1 2 1 − 1 2 β β lim (φ2′ f1 f1′ φ2)(r) = lim f1′ (r) r =0 if β =1. r 0+ − r 0+ 1 β 6 → → − by l’Hospital’s Rule. Moreover, in view of (3.40), (2.68) and (3.10), for r (0, t˜ ) ∈ 1 B(1 B) B(1 B) (φ′ f f ′ φ )′ = φ f − − + − 2 1 − 1 2 2 1 r2 (c + r)2 0
≤ Monotonicity implies that

β φ2′ f1′ (3.41) = on (0, t˜1) . r φ2 ≤ f1 Integrating (3.41) over [ǫ , t˜ ǫ ] (0, t˜ ) , and passing ǫ 0, we have 1 1 − 2 ⊂ 1 2 →

0 < C(ǫ )rβ f on [ǫ , t˜ ], 1 ≤ 1 1 1 where C(ǫ1) > 0 is a constant depending on ǫ1 . Thus f1 > 0 on (0, t˜1] . We claim t˜1 = . Otherwise there would exist a δ > 0 such that t˜1 + δ < at which f1 > 0 by the∞ continuity, and would lead to, via (3.10) a contradiction ∞

t˜ < t˜ + δ t˜ . 1 1 ≤ 1 B(1 B) B(1 B) Applying Theorem 2.8, in which G = − − = G , (c > 0) ,κ = 1 (c+r)2 ≥ (c+r)2 2 1 ′ f1 β ◦ 1 , , t˜1 = t1, t˜2 = t2, we have shown on Bt (x0) , (3.36) holds under (3.37) f1 ≥ r f 2 f for every c > 0. To prove (1) , we proceed analogously to the proof of Theorem ◦ 3.2 : For every x Bt (x ), there exist sequences xi in a radial geodesic ray in ∈ 2 0 { } ◦ Bt (x ) and ci > 0 such that xi x,ci 0 , and r(xi)+ ci = r(x) . It follows 1 0 { } → → that for every ci > 0 ,

1 1 B(1 B) B 2 + 2 K(r)(xi) − 2 ∆r(xi) (n 1)| − | , ≤ ci + r(xi) ⇒ ≥ − ci + r(xi) (3.42) and consequently , as i ,
→ ∞ B(1 B) B 1 + 1 K(r)(x) − (x) ∆r(x) (n 1)| − 2 | 2 , ≤ r2 ⇒ ≥ − r(x)

◦ and (3.36) holds on Bt2 (x0) . 

Corollary 3.4. If the radial curvature K satisfies B B 1 (3.43) K(r) resp.K(r) ,c 0 , where 0 B ≤ r2 ≤ (c + r)2 ≥ ≤ ≤ 4
DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS33

◦ on Bt (x ) D(x ), then 2 0 ⊂ 0 1+ √1 4B − Hess r g dr dr and 2r ≤ − ⊗ (3.44)   1+ √1 4B (n 1) − ∆r − 2r ≤

◦ hold pointwise on Bt2 (x0) .

2 1 √1 4b2 Proof. Let B(1 B) = b in (3.35). Then B = ± 2− , and by completing the square, we have− 1 1 1 b2 = B(1 B)= (B )2 + . − − − 2 4 ≤ 4 1 1 1+√1 4b2 Hence, B + = − . Substitute these into Theorem 3.4, so that this | − 2 | 2 2 Theorem is rephrased in terms of b2. We then replace b2 in this rephrased Theorem by B and obtain the desired. 

Theorem 3.5. If the radial curvature K satisfies B B 1 (3.45) 1 K(r) where 0 B B r2 ≤ ≤ r2 ≤ 1 ≤ ≤ 4

◦ on Bτ (x0) D(x0) , then (3.46) ⊂ 1+ √1 4B 1+ √1+4B − g dr dr Hess r 1 g dr dr and 2r − ⊗ ≤ ≤ 2r − ⊗     1+ √1 4B 1+ √1+4B (n 1) − ∆r (n 1) 1 − 2r ≤ ≤ − 2r

◦ hold pointwise on Bτ (x0). If in addition (3.33) occurs on D(x0) , then

1+ √1+4B (3.47) ∆r (n 1) 1 ≤ − 2r holds weakly on M. (2) Equivalently, if K satisfies B B 1 (3.48) 1 K(r) , 0 B B (c + r)2 ≤ ≤ (c + r)2 ≤ 1 ≤ ≤ 4

◦ ◦ on Bτ (x ) D(x ) , where c 0 , then (3.46) holds on Bτ (x ). If in addition, 0 ⊂ 0 ≥ 0 (3.33) occurs on D(x0) , then (3.47) holds weakly on M. Proof. This follows at once from Corollaries 3.3 and 3.4. 

Corollary 3.5. (1) Let the radial curvature K satisfy B (1 B ) B(1 B) (3.49) 1 − 1 K(r) − , 0 B,B 1 r2 ≤ ≤ r2 ≤ 1 ≤ 34 S.W. WEI

◦ on Bτ (x0) D(x0) . Then (3.50) ⊂ B 1 + 1 1+ 1+4B (1 B ) | − 2 | 2 g dr dr Hess r 1 − 1 g dr dr and r − ⊗ ≤ ≤ 2r − ⊗   p   B 1 + 1 1+ 1+4B (1 B ) (n 1)| − 2 | 2 ∆r (n 1) 1 − 1 − r ≤ ≤ − p 2r ◦ hold pointwise on Bτ (x0). If in addition, (3.25) occurs on D(x0) , then 1+ 1+4B (1 B ) (3.27) ∆r (n 1) 1 − 1 ≤ − p 2r holds weakly on M. (2) Equivalently, if K satisfies B (1 B ) B(1 B) (3.51) 1 − 1 K(r) − , 0 B,B 1 (c + r)2 ≤ ≤ (c + r)2 ≤ 1 ≤

◦ ◦ on Bτ (x ), where c 0 , then (3.50) holds on Bτ (x ). If in addition, (3.28) occurs 0 ≥ 0 on D(x0) , then (3.27) holds weakly on M. Corollary 3.6. If the radial curvature K satisfies (3.52) K(r) = 0(resp. 0 , 0) ≤ ≥ on M x , then \{ 0} 1 Hess r = (resp. Hess r , Hess r ) g dr dr and ≥ ≤ r − ⊗ (3.53)   1 ∆r = (resp. ∆r , ∆r ) (n 1) ≥ ≤ − r on M x . \{ 0} Proof. This follows at once from Corollary 3.5 in which B1 = B =1 , or Theorem 3.5, in which B1 = B =0 , or Corollary 3.2 in which A = 1(resp. A = 0). 

Corollary 3.7. If the radial curvature K satisfies A B K(r) −r2 ≤ ≤ r2 (3.54) A B 1 (resp. K(r) ) , 0 A, 0 B −(c + r)2 ≤ ≤ (c + r)2 ≤ ≤ ≤ 4 on M x , where c> 0 , then \{ 0} 1+ √1 4B 1+ √1+4A (3.55) − g dr dr Hess(r) g dr dr and 2r − ⊗ ≤ ≤ 2r − ⊗     1+ √1 4B 1+ √1+4A (3.55) (n 1) − ∆r (n 1) − 2r ≤ ≤ − 2r hold on M x . \{ 0} Proof. This follows at once from Theorems A and Corollary 3.4.  DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS35

4. Geometric Applications in Mean Curvature The following is an immediate geometric application:

Theorem 4.1. (Mean Curvature Comparison Theorems) Let H(r) be the mean curvature of the geodesic sphere ∂Br(x0) of radius r centered at x0 in M with respect to the unit outward normal. Then

A , if RicM (r) satisfies (2.39) or (2.41) , r rad (4.1) H(r)  ≤ 1+ 1+4B1(1 B1) M  − , if Ricrad(r) satisfies (2.51) or (2.53) ; p 2r   A , if K(r) satisfies (3.1) or (3.4) , r  1+ 1+4B1(1 B1) (4.2) H(r)  − , if K(r) satisfies (3.25) or (3.28) , ≤  2r  p 1+ √1+4B 1 , if K(r) satisfies (3.33) ;  2r    A 1 , if K(r) satisfies (3.5) or (3.7) , r  1 1  B 2 + 2 (4.3) H(r)  | − | , if K(r) satisfies (3.35) or (3.37) , ≥  r  1+ √1 4B − , if K(r) satisfies (3.43) ;  2r    (4.4) A A 1 H(r) , if K(r)satisfies (3.14) or (3.16) , r ≤ ≤ r  1+ √1+4A 1+ √1+4A  1 , H(r) (n 1) , if K(r) satisfies(3.17) or (3.21) ,   2r ≤ ≤ − 2r   1+ √1 4B 1+ √1+4B1  − , H(r) , if K(r) satisfies (3.45) or (3.48) ,  2r ≤ ≤ 2r  B 1 + 1 1+ 1+4B (1 B ) | − 2 | 2 H(r) 1 − 1 , if K(r) satisfies (3.49) or (3.51) ,  r ≤ ≤ 2r  p  1+ √1 4B 1+ √1+4A  − H(r) , if K(r) satisfies (3.54) ;  2r ≤ ≤ 2r    A 1+ √1+4A = , resp. , if K(r) satisfies (3.22) or (3.24) , r 2r (4.5) H(r)  1
 = (resp. , ) , if K(r) satisfies (3.52) ; ≥ ≤ r  where the corresponding geodesic spheres ∂Br(x0) are as in Theorems 3.1-3.5, The- orem A, and Corollaries 3.1-3.7. 36 S.W. WEI

Proof. By Gauss lemma (4.6) 1 1 ∆r := trace Hess r n 1 n 1 − − 1
= the trace of the second fundamental form of ∂B (x ) in M n 1 r 0 − := H(r)
(see [38, (3.28)]). The results follow from Theorems 2.6-2.7, 3.1-3.5, Theorem A, and Corollaries 3.1-3.7. 

5. The Growth of Bundle-Valued Differential Forms and Their Interrelationship Let (M,g) be a smooth Riemannian manifold. Let ξ : E M be a smooth Riemannian vector bundle over (M,g) , i.e. a vector bundle such→ that at each fiber k k is equipped with a positive inner product , E . Set A (ξ) = Γ(Λ T ∗M E) the space of smooth k forms on M with valuesh ini the vector bundle ξ : E ⊗M. − k → For two forms Ω, Ω′ A (ξ), the induced inner product Ω, Ω′ is defined as in k ∈ 2 q h q i (6.4). For Ω A (ξ), set Ω = Ω, Ω . Then Ω = Ω, Ω 2 and we are ready to make the following∈ | | h i | | h i Definition 5.1. For a given q R , a function or a differential form or a bundle- valued differential form f has p∈-finite growth (or, simply, is p-finite) if there exists x M such that 0 ∈ 1 (5.1) lim inf f q dv < , r rp | | ∞ →∞ ZB(x0;r) and has p-infinite growth (or, simply, is p-infinite) otherwise. For a given q R , a function or a differential form or a bundle-valued differential form f has p-mild∈ growth (or, simply, is p-mild) if there exist x M, and a strictly 0 ∈ increasing sequence of rj ∞ going to infinity, such that for every l > 0, we have { }0 0 1 p p−1 ∞ (rj rj ) (5.2) +1− R f q dv = , B(x0;rj+1)\B(x0;rj ) ∞ j=ℓ0  | |  P and has p-severe growth (or, simply, is p-severe) otherwise. For a given q R , a function or a differential form or a bundle-valued differential ∈ form f has p-obtuse growth (or, simply, is p-obtuse) if there exists x0 M such that for every a> 0, we have ∈

1 p−1 (5.3) ∞ 1 a R f qds dr = , ∂B(x0;r) ∞  | |  and has p-acute growth (Ror, simply, is p-acute) otherwise. For a given q R , a function or a differential form or a bundle-valued differential ∈ form f has p-moderate growth (or, simply, is p-moderate) if there exist x0 M, and ψ(r) , such that ∈ ∈F 1 q (5.4) lim sup p p 1 f dv < , r r ψ − (r) B(x ;r) | | ∞ →∞ Z 0 DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS37 and has p-immoderate growth (or, simply, is p-immoderate) otherwise, where

∞ dr (5.5) = ψ :[a, ) (0, ) = for some a 0 . F { ∞ −→ ∞ | rψ(r) ∞ ≥ } Za (Notice that the functions in are not necessarily monotone.) F For a given q R , a function or a differential form or a bundle-valued differential ∈ form f has p-small growth (or, simply, is p-small) if there exists x0 M, such that for every a> 0 ,we have ∈ 1 p−1 (5.6) ∞ r a R f q dv dr = , B(x0;r) ∞  | |  and has p-large growth (or,R simply, is p-large) otherwise. Definition 5.2. For a given q R , a function or a differential form or a bundle- valued differential form f has p-balanced∈ growth (or, simply, is p-balanced) if f has one of the following: p-finite, p-mild, p-obtuse, p-moderate, or p-small growth, and has p-imbalanced growth ( or, simply, is p-imbalanced) otherwise. The above definitions of “p-balanced, p-finite, p-mild, p-obtuse, p-moderate, p-small” and their counter-parts “p-imbalanced, p-infinite, p-severe, p-acute, p- immoderate, p-large” growth depend on q, and q will be specified in the context in which the definition is used. Theorem 5.1. For a given q R , f is p-moderate(resp. p-immoderate) if and only if f is p-small(resp. p-large)∈ . Proof. ( ) Since f has p-moderate growth, we may assume, by the definition of the limit⇒ superior of functions, there exists a ψ as in (5.5) , and a constant 1 q ∈ F K′ > 0 such that p p−1 f dv K′ for r>ℓ0 . This implies that r ψ (r) B(x0;r) | | ≤
r R 1 1 (5.7) q p 1 p 1 for r>ℓ0 . f dv ≥ K′ r − ψ − (r) B(x0;r) | | 1 Taking both sidesR to the power p 1 and integrating, we have − 1 ∞ r p−1 1 ∞ 1 (5.8) q dr 1 dr for r>ℓ0 . a f dv ≥ (K ) p−1 a rψ(r) Z  B(x0;r) | |  ′ Z If f were p-large,R (5.8) would lead to ∞ 1 dr < , contradicting (5.5). a rψ(r) ∞ ( ) If f q dv = 0 for r>a> 0, then we are done. Or, we let ψ(r) = B(x0;r) R ⇐ | | 1 1 q p−1 ( p R f dv) , r > a . Then r B(x0;r) | | 1 R 1 r p−1 (5.9) = . rψ(r) f q dv  B(x0;r) | |  Integrating (5.9) from a to , we haveR ∞ 1 ∞ 1 ∞ r p−1 dr = q dr = , a rψ(r) a f dv ∞ Z Z  B(x0;r) | |  by the assumption of f being p-small.R Hence ψ(r) . Suppose the contrary, f were p-immoderate, i.e. (5.4) were not true. Then we∈ wouldF have, via (5.9) 38 S.W. WEI

1 q lim sup p p 1 f dv = , contradicting r r ψ − (r) B(x ;r) | | ∞ →∞ Z 0 1 q 1 r q lim sup p p−1 f dv = lim sup q f dv =1 .  r ψ (r) B(x0;r) r R f dv B(x0;r) r | | r B(x0;r) | | | | →∞ →∞   R R Theorem 5.2. For a given q R , if f has p-small growth then f has p-mild growth. ∈

Proof. For a strictly increasing sequence rj with rj+1 =2rj , we obtain (5.10) { } 1 1 ℓ p p−1 ℓ p−1 (rj+1 rj ) rj+1 rj p 1 − q − q (rj+1 rj ) − RB(x ;r )\B(x ;r ) f dv RB(x ;r ) f dv j=ℓ0 0 j+1 0 j | | ≥ j=ℓ0 0 j+1 | | · −    1  P Pℓ p−1 rj+1 rj = − q (rj+1 rj ) RB(x ;r ) f dv j=ℓ0 0 j+1 | | · −   1 Pℓ 1 p−1 rj+2 22 1 = q 2 (rj+2 rj+1) RB(x ;r ) f dv j=ℓ0 0 j+1 | | · −   1 Pℓ p−1 1 rj+2 r p+1 r f q dv dr p−1 j+1 RB(x ;r) ≥ j=ℓ0 2 0 | |  1 P R p−1 1 rℓ+2 r = p+1 r f q dv dr . p−1 ℓ0+1 RB(x ;r) 2  0 | |  R where we have applied the Mean Value Theorem for integrals to the forth step. Now suppose contrary, f were p-severe. Letting l in (5.10) would imply → ∞ ℓ 1 1 p p−1 p−1 (rj+1 rj ) 1 ∞ r > − q p+1 q dr , ∞ f dv ≥ p−1 r f dv j ℓ B(x0;rj+1) B(x0;rj ) 2 ℓ0+1 B(x0;r) X= 0  \ | |  Z  | |  contradictingR the hypothesis that f has a p-small growth. R 

Theorem 5.3. For a given q R , ∈ (i) If f is p-acute (resp. p-mild) then f is p-severe (resp. p-obtuse). If f is p-severe (resp. p-moderate or p-small ) then f is p-immoderate and p-large (resp. p-mild). (ii) Conversely, suppose f qdv is convex in r, then f is p-immoderate or B(x0;r) | | p-large implies that f is p-severe or p-acute. Hence, f is p-severe, p-acute, R p-immoderate, and p-large are all equivalent.

Proof. (i) 0

Hence, for every strictly increasing sequence of rj and every rℓ > a, setting { } 0 s = ri,t = ri and summing over i from ℓ to ℓ and letting ℓ , one gets: +1 0 → ∞ 1 p p−1 1 ∞ (ri+1 ri) 1 p−1 (5.12) − q ∞ d q dr . 2 2 a R f R (f +ǫ) dv ≤ dr B(x0;r) i=ℓ0  B(x0;ri+1)\B(x0;s)  | | P R
Let ǫ 0, we get the desired first assertion by the coarea formula and the Domi- nated→ Convergence Theorem. The proof of the second assertion follows from The- orem 5.1 and Theorem 5.2. (ii) By the convexity assumption, comparing the slope of the tangent and the slope of the chord at two distinct points (analogous to [22]), we have

q q d B x r f dv B x f dv r> 0 f qdv ( 0; ) | | − ( 0;0) | | a. e.. ⇒ dr | | ≥ r 0 ZB(x0;r) R − R 1 1 p r p Hence, ∞ 1 −1 dr ∞ −1 dr < . The assertion a d R f q dv a R f q dv dr B(x0;r) | | ≤ B(x0;r) | | ∞ follows fromR the coarea formula.
R


Proof of Theorem 5.4. This follows at once from Theorems 5.1, 5.2 and 5.3. 

Definition 5.3. A function or differential form or bundle-valued differential form f on M is said to belong to Lp(M) (or, simply, is Lp) , denoted by f Lp(M) if f pdv < . ∈ M | | ∞ CorollaryR 5.1. Every Lq function or differential form or bundle-valued differential form f on M has p-balanced growth, p 0 , and in fact, has p-finite, p-mild, p- obtuse, p-moderate, and p-small growth,≥p 0 , for the same value of q ≥ Proof. Since f is Lq on M, by Definition 5.3, we may assume f q dv = K˜ for M | | some constant K˜ < . It follows from Definition 5.1 that every Lq function or R differential form or bundle-valued∞ differential form f on M is both p-finite and p-small, p 0 . Indeed, ≥ 1 1 lim inf f q dv lim inf K˜ =0 < r rp | | ≤ r rp ∞ →∞ ZB(x0;r) →∞ 1 1 ∞ r p−1 ∞ r p−1 and q dr = . a f dv ≥ a K˜ ∞ Z  B(x0;r) | |  Z   Now the assertion followsR from Definition 5.2 and Theorem 5.4. 

6. Generalized harmonic forms with values in vector bundle, decomposition, integrability and growth By Stokes’ theorem, a smooth differential form ω on a compact Riemannian manifold is harmonic if and only if it is closed and co-closed. That is,

(6.1) ∆ω = 0 if and only if dω = 0 and d⋆ω =0. 40 S.W. WEI

k k+1 The exterior differential operator d∇ : A (ξ) A (ξ) relative to the connec- tion E is given by → ∇ k+1 i+1 E d∇σ (X , ..., Xk )= ( 1) (σ(X , ..., Xi, ..., Xk )) 1 +1 − ∇Xi 1 +1 i=1 (6.2) X i+j + ( 1) σ([Xi,Xj],Xb, ..., Xi, ..., Xj , ..., Xk ) , − 1 +1 i

Relative to the Riemannian structures of E and TM, the codifferential operator k k 1 δ∇ : A (ξ) A − (ξ) is characterized as the adjoint of d via the formula →

d∇σ, ρ dvg = σ, δ∇ρ dvg , h i h i ZM ZM k 1 k where σ A − (ξ),ρ A (ξ) , one of which has compact support, and dvg is the volume element∈ associated∈ with the metric g on TM. Then

(6.6) (δ∇ρ)(X1, ..., Xk 1)= ( ei ρ)(ei,X1, ..., Xk 1) . − − ∇ − i X k k Let ∆∇ : A (ξ) A (ξ) be given by → (6.7) ∆∇ = (d∇δ∇ + δ∇d∇) . − Definition 6.2. Ω Ak(ξ) is said to be a harmonic form with values in the vector ∈ bundle ξ : E M, if ∆∇Ω=0, and a generalized harmonic form with values in → the vector bundle ξ : E M, if Ω, ∆∇Ω 0. → h i≥ Theorem 6.1. (Duality Theorem) (i) Ω Ak(ξ) is harmonic if and only if n k k ∈ ⋆ Ω A − (ξ) is harmonic. (ii) Ω A (ξ) has 2-balanced growth on M, for q =2∈, or for 1 < q(= 2) < 3 with Ω satisfying∈ Condition W see (6.5) , if and only n k 6 if ⋆ Ω A − (ξ) has 2-balanced growth on M, for q =2 , or for 1 < q(= 2) < 3 with ∈ 6
DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS41

⋆ Ω satisfying Condition W see (6.5) . (iii) ⋆ Ω is a solution of ⋆ Ω, ∆ ⋆ Ω 0 on M if and only if ⋆ Ω is closed and co-closed. h i≥
Theorem 6.2. (Unity Theorem) If a bundle-valued differential k-form Ω Ak(ξ) has 2-balanced growth, for q = 2 , or for 1 < q(= 2) < 3 with Ω satisfying∈ 6 Condition W, then the following six statements (i) Ω, ∆∇Ω 0 , (ii) ∆∇Ω=0 , ∇ h i ≥ (iii) d∇ Ω = δ Ω=0 , (iv) ⋆ Ω, ∆∇ ⋆ Ω 0 , (v) ∆∇ ⋆ Ω=0 , (vi) d∇ ⋆ Ω = h i ≥ δ∇ ⋆ Ω=0 are equivalent. Proof. (i) (ii) (iii) It is obvious. (i) (iii) Choose a smooth cut- off function⇐ψ(x) as⇐ in [35, (3.1)], i.e. for any x ⇒ M and any pair of positive
0
numbers s,t with s < t , a rotationally symmetric Lipschitz∈ continuous nonnegative function ψ(x) = ψ(x; s,t) satisfies ψ 1 on B(s), ψ 0 off B(t) , and ψ C1 ≡ ≡ |∇ | ≤ t s , a.e. on M , where C1 > 0 is a constant (independent of x0,s,t). Let 1− 0 , we compute ∞ 2 2 q 1 (6.8) 0 ψ ( Ω + ǫ) 2 − Ω, ∆∇Ω dv ≤ B(t)h | | i and obtain R

2 2 q 1 2 2 2 ψ ( Ω + ǫ) 2 − ( d∇Ω + δ∇Ω ) dv B(t) | | | | | | q 2√2C1 2 1 2 R ( Ω + ǫ) 2 dv
(6.9) ≤ 1 q 2 (t s)2 B(t) B(s) | |  −| − | − \ 
2 R 2 q 1 2 2 ψ ( Ω + ǫ) 2 − ( d∇Ω + δ∇Ω ) dv . · B(t) B(s) | | | | | |  \  R Proceeding as in [39], we obtain the desired (iii). Hence, (i) (ii) (iii). On the other hand, applying Duality Theorem 6.1, we prove (ii) (v)⇔ , and⇔ (iv) (v) (vi) . Consequently, (i) through (vi) are all equivalent. ⇔ ⇔ ⇔

Corollary 6.1. Every L2 bundle-valued harmonic k-form Ω Ak(ξ) is closed and co-closed and every Lq, 1 < q(= 2) < 3 bundle-valued harmonic∈ k-form Ω Ak(ξ) satisfying Condition W is closed6 and co-closed. ∈

7. Applications in Geometric Differential-Integral Inequalities on manifolds The derived Laplacian comparison Theorems 2.6 - 2.7 and Hessian comparison Theorems 3.1 - 3.4 have many applications. In particular, they shed light on geo- metric differential-integral inequalities on Riemannian manifolds. Using an analog of Bochner’s Method or “B2 4AC 0” Method, Wei and Li [41] proved the following theorem. − ≤ Theorem 7.1. ([41]) For every u W 1,2(M x ), and every a,b R , the ∈ 0 \{ 0} ∈ following inequality holds on a manifold M with a pole x0 :

1 1 2 2 1 u 2 u 2 u 2 (7.1) | | (r∆r a b)dv | | dv |∇ | dv , 2 ra+b+1 − − ≤ r2a r2b M M  M  R R R where dv is the volume element of M. 42 S.W. WEI

7.1. Generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds. R 1,p 1,p Definition 7.1. u : M is said to belong to W0 (M) (or, simply, is W0 ) 1,p → , denoted by u W0 (M) if there exists a sequence ui in C0∞(M) such that ∈ 1 { } p p p u ui + (u ui) dv 0, as i . M | − | |∇ − | → → ∞ R Applying Corollary 3.6 to Theorem
7.1, Wei and Li [41] proved the following theorem. Theorem 7.2. ([41]) Let M be an n-dimensional complete Riemannian manifold with a pole such that the radial curvature K(r) of M satisfies one of the following three conditions:

(i)K(r) 0 and n a + b +1, ≥ ≤ (7.2) (ii)K(r) 0 and a + b +1 n, ≤ ≤ (iii)K(r) = 0 and a,b R are any constants . ∈ Then for every u W 1,2(M x ), and a,b R ∈ 0 \{ 0} ∈

1 1 2 2 u 2 u 2 u 2 (7.3) C | | dv | | dv |∇ | dv , ra+b+1 ≤  r2a   r2b  ZM ZM ZM     where the constant C is given by

n (a+b+1) − 2 , if K(r) satisfies (i), −n (a+b+1) − , if K(r) satisfies (ii), (7.4) C = C(a,b)=  2  n (a+b+1)  − 2 , if K(r) satisfies(iii) .

 The case M = Rn is due to Caffarelli et al. [7] and Costa [14].  Theorem 7.3. (Generalized sharp Caffarelli-Kohn-Nirenberg type inequalities un- der radial Ricci curvature assmumptions) Let M be an n-dimensional complete M Riemannian manifold with a pole such that the radial Ricci curvature Ricrad of M satisfies one of the following five conditions:

(7.5) A(A 1) a + b + A (i)RicM (r) (n 1) − and n , rad ≥− − r2 ≤ A A(A 1) a + b + A (ii)RicM (r) (n 1) − and n , rad ≥− − (c + r)2 ≤ A

M B1(1 B1) 2a +2b +1+ 1+4B1(1 B1) (iii)Ricrad(r) (n 1) − and n − , ≥ − r2 ≤ 1+ 1+4B (1 B ) p 1 − 1 M B1(1 B1) 2a +2b +1+p 1+4B1(1 B1) (iv)Ricrad(r) (n 1) − and n − , ≥ − (c + r)2 ≤ 1+ 1+4B (1 B ) p 1 − 1 (v)RicM (r) 0 and n a + b +1 , rad ≥ ≤ p DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS43

1,2 where 0 B1 1 A are constants. Then for every u W0 (M x0 ), and a,b R ≤ ≤ ≤ ∈ \{ } ∈

1 1 2 2 u 2 u 2 u 2 (7.3) C | | dv | | dv |∇ | dv , ra+b+1 ≤  r2a   r2b  ZM ZM ZM    

where

C = C(a,b,A,B1) a+b (n 1)A M − − , if Ric satisfies (i) or (ii), (7.6) 2 rad 2a+2b (n 1) 1+√1+4B1(1 B1) = − − − M  4 , if Ricrad satisfies (iii) or (iv),  a+b+1 n M  2 − ,
if Ricrad satisfies (v). 

Proof. This follows from Theorem2.6 and Theorem 2.7 that we can apply (2.40) and (2.52) to simplify the integral inequality (7.1), based on Theorem 7.1 and obtain the desired. 

Theorem 7.4. (Generalized sharp Caffarelli-Kohn-Nirenberg type inequalities un- der radial curvature assmumptions) Let M be an n-dimensional complete Riemann- ian manifold with a pole such that the radial curvature K(r) of M satisfies one of the following twelve conditions: 44 S.W. WEI

(7.7) A(A 1) a + b + A (i)K(r) − and n , ≥− r2 ≤ A A(A 1) a + b + A (ii)K(r) − and n , ≥− (c + r)2 ≤ A

A1(A1 1) a + b + A1 (iii)K(r) 2 − and n , ≤− r ≥ A1 A1(A1 1) a + b + A1 (iv)K(r) −2 and n , ≤− (c + r) ≥ A1 A(A 1) (v)K(r)= − and a,b R are any constants, − r2 ∈ A(A 1) (vi)K(r)= − and a,b R are any constants, − (c + r)2 ∈ A (vii)K(r)= and a,b R are any constants, −r2 ∈ A (viii)K(r)= and a,b R are any constants, −(c + r)2 ∈

B (1 B ) 2a +2b +1+ 1+4B1(1 B1) (ix)K(r) 1 − 1 and n − , ≥ r2 ≤ 1+ 1+4B (1 B ) p 1 − 1 B1(1 B1) 2a +2b +1+p 1+4B1(1 B1) (x)K(r) − and n − , ≥ (c + r)2 ≤ 1+ 1+4B (1 B ) p 1 − 1 B(1 B) a + b + B 1 + 1 (xi)K(r) − and n | p− 2 | 2 , ≤ r2 ≥ B 1 + 1 | − 2 | 2 B(1 B) a + b + B 1 + 1 (xii)K(r) − and n | − 2 | 2 , ≤ (c + r)2 ≥ B 1 + 1 | − 2 | 2 where 0 B,B1 1 A, and 0 c are constants. Then for every u 1,2 ≤ ≤ ≤ ≤ ∈ W (M x ), and a,b R 0 \{ 0} ∈

1 1 2 2 u 2 u 2 u 2 (7.3) C | | dv | | dv |∇ | dv , ra+b+1 ≤  r2a   r2b  ZM ZM ZM     where the constant C is given by (7.8) a+b (n 1)A − 2 − , if K(r) satisfies (i) or (ii), a+b (n 1)A − 2 − , if K(r) satisfies (iii) or (iv),  −a+b (n 1)A  − 2 − , if K(r) satisfies (v) or (vi),  C = C(a,b,B)=  2a+2b (n 1) 1+√1+4A  − − , if K(r) satisfies (vii) or (viii),  4 2a+2b (n 1) 1+√ 1+4B1(1
B1) − − − 4 , if K(r) satisfies (ix) or (x),  1 1  (n 1)( B 2 + 2 ) a b  − | − | − − ,
if K(r) satisfies (xi) or (xii).  2   DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS45

Proof. This follows from Theorems 3.1, 3.2, Corollary 3.2, and Theorems 3.3, 3.4 that we can apply (3.3), (3.6), (3.23) , (3.27), and (3.36) to simplify the integral inequality (7.1), based on Theorem 7.1.  7.2. Embedding theorems for weighted Sobolev spaces of functions. Hes- sian comparison theorems via these geometric inequalities lead to embedding the- orems for weighted Sobolev spaces of functions on Riemannian manifolds. Let M be manifold as in Theorem 7.3, or Theorem 7.4, or Theorem 7.2. Following Costa’s 1,2 notation [14], or [41], we let Dγ (M) denote the completion of C0∞(M x0 ) with respect to the norm \{ }

1 2 u 2 (7.9) u 1,2 := |∇ | dv . || ||Dγ (M)  r2γ  ZM 2   L (M) denote the completion of C∞(M x ) with respect to the norm γ 0 \{ 0}

1 2 u 2 (7.10) u L2 M := | | dv || || γ ( )  r2γ  ZM 1   and Ha,b(M) denote the completion of C0∞(M x0 ) with respect to the Sobolev norm \{ }

2 2 1 u u 2 (7.11) u H1 (M) := | | + |∇ | dv . || || a,b r2a r2b Z  M
1,2 1,2 If γ = 0 in (7.9), we simply write D0 (M) = D (M) . Applying arithmetic- mean-geometric-mean inequality to Theorems 7.3 - 7.4 and 7.2, we have Theorem 7.5. (Embedding Theorem) Let M be a manifold of radial curvature K satisfying one of the five conditions in (7.5), or one of the twelve conditions in (7.7), or one of the three conditions in (7.2) . Then the following continuous embeddings hold

1 2 1 2 (7.12) Ha,b(M) L a+b+1 (M) and Hb,a(M) L a+b+1 (M) . ⊂ 2 ⊂ 2 7.3. Geometric differential-integral inequalities on manifolds. As a conse- quence of embedding theorem, we have geometric differential-integral inequalities on manifolds: Theorem 7.6. Let M be a manifold of radial curvature K satisfying one of the five conditions in (7.5), or one of the twelve conditions in (7.7). Then we have the following seven geometric differential-integral inequalities: i) For any u H1 (M) , ∈ b+1,b

u 2 u 2 (7.13) C | | dv |∇ | dv , 1 r2(b+1) ≤ r2b ZM ZM 46 S.W. WEI where (7.14) 2 (n 1)A 1 M − − b , if Ric (r) satisfies (7.5)(i) or (7.5)(ii) 2 − rad    or K(r)satisfies any of (7.7)(i) (vi) ,  2 −  4b+2 (n 1) 1+√1+4A  − − , if K(r) satisfies (7.5)(vii) or (7.5)(viii) ,  4   
 2 C1 =  4b+2 (n 1) 1+√1+4B1(1 B1)  − − − , if RicM (r) satisfies (7.5)(iii) or (7.5)(iv)  4 rad 
  or K(r) satisfies (7.7)(ix) or (7.7)(x) ,  2b+2 n 2 M  2 − , if Ricrad(r) satisfies (7.5)(v) ,  2  (n 1) B 1 + n−3 2b  − | − 2 |
2 − , if K(r) satisfies (7.7)(xi) or (7.7)(xii),  2      in which a = b +1 for (7.5) and (7.7); ii) For any u H1 (M) , ∈ a,a+1

2 u 2 u 2 u 2 (7.15) C | | dv | | dv |∇ | dv , 2  r2(a+1)  ≤  r2a   r2(a+1)  ZM ZM ZM       where (7.16) 2 (n 1)A 1 M − − a , if Ric (r) satisfies (7.5)(i) or (7.5)(ii) 2 − rad    or K(r)satisfies any of (7.7)(i) (vi) ,  2 −  4a+2 (n 1) 1+√1+4A  − − , if K(r) satisfies (7.5)(vii) or (7.5)(viii) ,  4   
 2 C2 =  4a+2 (n 1) 1+√1+4B1(1 B1)  − − − , if RicM (r) satisfies (7.5)(iii) or (7.5)(iv)  4 rad 
  or K(r) satisfies (7.7)(ix) or (7.7)(x) ,  2a+2 n 2 M  2 − , if Ricrad(r) satisfies (7.5)(v) ,  2  (n 1) B 1 + n−3 2a  − | − 2 |
2 − , if K(r) satisfies (7.7)(xi) or (7.7)(xii),  2      in which b = a +1 for (7.5) and (7.7) ; iii 1 2 ) If u H (b+1),b(M) then u L (M) and ∈ − ∈

2 u 2 (7.17) C u 2 dv r2(b+1) u 2 dv |∇ | dv , 3  | |  ≤  | |   r2b  ZM ZM ZM       DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS47 where (7.18) 2 (n 1)A+1 M − 2 , if Ricrad(r) satisfies (7.5)(i) or (7.5)(ii)    or K(r)satisfies any of (7.7)(i) (vi) ,  2 −  2+(n 1) 1+√1+4A  − , if K(r) satisfies (7.5)(vii) or (7.5)(viii) ,  4   
 2 C3 =  2+(n 1) 1+√1+4B1(1 B1)  − − , if RicM (r) satisfies (7.5)(iii) or (7.5)(iv)  4 rad 
  or K(r) satisfies (7.7)(ix) or (7.7)(x) ,  n2 M  4 , if Ricrad(r) satisfies (7.5)(v) ,  2  (n 1) B 1 + n+1  − | − 2 | 2 , if K(r) satisfies (7.7)(xi) or (7.7)(xii),  2      in which a = b 1 for (7.5) and (7.7); iv) If u −H1−(M), then u L2(M) and ∈ 0,1 ∈ 1

2 u 2 u 2 (7.19) C | | dv u 2 dv |∇ | dv , 4  r2  ≤  | |   r2  ZM ZM ZM       where (7.20) 2 (n 1)A 1 M − 2 − , if Ricrad(r) satisfies (7.5)(i) or (7.5)(ii)    or K(r)satisfies any of (7.7)(i) (vi) , 2 −  2 (n 1) 1+√1+4A  − −  4 , if K(r) satisfies (7.5)(vii) or (7.5)(viii) ,   
 2 C4 =  (n 1) 1+√1+4B1(1 B1) 2  − − − , if RicM (r) satisfies (7.5)(iii) or (7.5)(iv)  4 rad 
  or K(r) satisfies (7.7)(ix) or (7.7)(x) ,  n 2 2 M  ( −2 ) , if Ricrad(r) satisfies (7.5)(v) ,  2  (n 1) B 1 + n−3  − | − 2 | 2  2 , if K(r) satisfies (7.7)(xi) or (7.7)(xii),      in which a =0 ,b =1 for (7.5) and (7.7) ; v 1 2 ) If u H 1,1(M), then u L 1 (M) and ∈ − ∈ 2

2 u 2 u 2 (7.21) C | | dv r2 u 2 dv |∇ | dv , 5  r  ≤  | |   r2  ZM ZM ZM       48 S.W. WEI where (7.22) 2 (n 1)A M −2 , if Ricrad(r) satisfies (7.5)(i) or (7.5)(ii)    or K(r)satisfies any of (7.7)(i) (vi) ,  2 −  (n 1) 1+√1+4A  −  4 , if K(r) satisfies (7.5)(vii) or (7.5)(viii) ,   
 2 C5 =  (n 1) 1+√1+4B1(1 B1) M  − − , if Ric (r) satisfies (7.5)(iii) or (7.5)(iv)  4 rad
  or K(r) satisfies any of (7.7)(ix) or (7.7)(x) ,  n 1 2 M  ( − ) , if Ric (r) satisfies (7.5)(v) ,  2 rad  (n 1)( B 1 + 1 ) 2  − | − 2 | 2 , if K(r) satisfies (7.7)(xi) or (7.7)(xii) ,  2     in which a = 1 ,b =1 for (7.5) and (7.7); vi −1 1 2 ) If u H (M)= H0,0(M), then u L 1 (M) and ∈ ∈ 2 2 u 2 (7.23) C | | dv u 2 dv u 2 dv , 6  r  ≤  | |   |∇ |  ZM ZM ZM where      

(7.24) C6 = C5 , in which a =0 ,b =0 for (7.5) and (7.7) ; vii) For any u D1,2(M), ∈ u 2 (7.25) C | | dv u 2 dv , 7 r2 ≤ |∇ | ZM ZM where (7.26) 2 (n 1)A 1 M − 2 − , if Ricrad(r) satisfies (7.5)(i) or (7.5)(ii)    or K(r)satisfies any of (7.7)(i) (vi) , 2 −  2 (n 1) 1+√1+4A  − −  4 , if K(r) satisfies (7.5)(vii) or (7.5)(viii) ,   
 2 C7 =  (n 1) 1+√1+4B1(1 B1) 2  − − − , if RicM (r) satisfies (7.5)(iii) or (7.5)(iv)  4 rad 
  or K(r) satisfies (7.7)(ix) or (7.7)(x) ,  n 2 2 M  ( −2 ) , if Ricrad(r) satisfies (7.5)(v) ,  2  (n 1) B 1 + n−3  − | − 2 | 2 , if K(r) satisfies (7.7)(xi) or (7.7)(xii),  2     in which a =1 ,b =0 for (7.5) and (7.7). Remark 7.1. The case M = Rn is due to Costa ([14]). Proof. We make special choices in Theorems 7.3 and 7.4 as follows: i) Let a = b + 1; ii) Let b = a + 1; iii) Let a = b 1; − − DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS49

iv) Let a =0,b = 1; v) Let a = 1,b = 1; vi) Let a =0−,b = 0; vii) Let a =1,b =0. 

7.4. Generalized sharp Hardy type inequalities on Riemannian manifolds. Laplacian comparison Theorems further lead to Theorem 7.7. (Generalized Sharp Hardy Type Inequality) Let M be an n-manifold with a pole satisfying A(A 1) (2.39) RicM (r) (n 1) − where A 1 . rad ≥− − r2 ≥ Then for every u W 1,p(M), u Lp(M) with p> (n 1)A +1, we have ∈ 0 r ∈ − p p 1 (n 1)A p u (7.27) − − − | | dv u p dv . p rp ≤ |∇ | ZM ZM
We will apply a double limiting arguement to the following Lemma 7.1. (Geometric Differential-Integral Inequality) (see [41, (1.3)], [13, (3)], 1 [42, (8.1)]) Let u C∞(M) and ∂Bδ(x ) be the C boundary of the geodesic ball ∈ 0 0 Bδ(x0) centered at x0 with radius δ > 0. Let V be an open set with smooth boundary ∂V such that V M, and u =0 off V . We choose a sufficiently small δ > 0 so ⊂⊂ that ∂V ∂Bδ(x )= . Then for every ǫ> 0 and p> 1, we have ∩ 0 ∅ p p r p (r +ǫ)(r∆r+1) pr p p u dS + p 2 − u dv V ∂Bδ (x0) r +ǫ M Bδ (x0) (r +ǫ) − ∩ | | \ p−1 | |1 p−1 p p p (7.28) R u r p−1R p p | r|p ǫ dv u dv , ≤ M Bδ (x0) + M Bδ (x0) |∇ |  \   \  R
R where dS and dv are the volume element of ∂Bδ(x0) and M respectively. Proof of Theorem 7.1. Applying the Laplacian comparison Theorem 2.6 under the A(A 1) assumption Ricrad (n 1) r2− , one has r∆r +1 (n 1)A +1

For sufficiently small δ > 0, one has r (7.30) u pdS = 0 if x / V rp + ǫ| | 0 ∈ Z∂Bδ (x0) and

r p (7.31) p u dS 0 as δ 0 if x0 V, ∂Bδ (x0) r + ǫ| | → → ∈ Z

50 S.W. WEI

r Indeed, rp+ǫ is a continuous, nondecreasing function for r [0,δ0], where δ0 = 1 ∈ ǫ p ( p 1 ) and u is bounded in M. Hence, for δ<δ0 , − r δ (7.32) u pdS 6 max u pdS. p p M ∂Bδ (x0) r + ǫ| | δ + ǫ ∂Bδ (x0) | | Z Z

This implies (7.31). It follows from (7.28), via (7.30) or (7.31) that for every ǫ> 0, p (n 1)A 1 rp (n 1)A +1 ǫ − − − − − u p dv (rp + ǫ)2 | | M Bδ (x0)

(7.33) Z \ p 1 p−1 1 − p p p u r p−1 p p | p| dv u dv . ≤ M B (x ) r + ǫ M |∇ |  Z \ δ 0   Z 
We observe that the integrands in the left, and in the first factor in the right of (7.33) are monotone and uniformly bounded above by a positive constant multiple u p u p of r on M. Since r L (M), by the dominated convergent theorem, as ǫ 0 , (7.34) ∈ → p−1 p 1 u p u p p p (n 1)A 1 dv 6 p dv u p dv . r r − − − M Bδ (x0) ! M Bδ (x0) ! M |∇ | Z \ Z \ Z 
Simplifying and raising it to the p-th power,

p (n 1)A 1 p u p (7.35) − − − dv 6 u pdv. p M r M |∇ |   Z  Z 1,p Now we extend (7.35) from u C0∞(M) to u W0 (M) . Let ui be a sequence ∈ 1,p ∈ { } of functions in C∞(M) tending to u W (M) . Applying the inequality (7.35) to 0 ∈ 0 difference ui ui , we have m − n

1 1 p p p ui ui p (7.36) m − n dv 6 u pdv . r p (n 1)A 1 |∇ | ZM    ZM  − − − Hence ui is a Cauchy sequence in Lp(M) . Now we recall the following { r } Lemma 7.2. (see Theorem 3.3 in [42]) Lp(M) , 1 p is complete, i.e. p ≤ ≤ ∞ every Cauchy sequence ui in L (M) converges This means that if for every { } 1 p p ǫ > 0, there exists N such that M ui uj dv < ǫ, when i>N and j>N, | − | 1 p p p then there exists a unique function u L (M), such that ui u dv 0, R ∈
M | − | → as i . → ∞ R
p Lemma 7.3.
(see Theorem 3.4 in [42]) If ui is a Cauchy sequence in L (M) , 1 { } ≤ p , then there exists a subsequence uik and a nonnegative function U in Lp≤(M ∞) such that { } (1) ui 6 U almost everywhere in M. | k | (2) limk ui = u almost everywhere in M. →∞ k In view of Lemma 7.2, there exists a limiting function f(x) Lp(M) satisfying ∈ DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS51

(7.37) p p p ui(x) p p f(x) dv = lim | | dv 6 lim ui dv | | i rp p (n 1)A 1 i |∇ | ZM →∞ ZM  − − −  →∞ ZM p p 6 u p dv. p (n 1)A 1 |∇ |  − − −  ZM

1 On the other hand, since p is bounded in M Bǫ(x ) , where Bǫ(x ) is the open r \ 0 0 geodesic ball of radius ǫ > 0 , centered at x0, and the pointwise convergence in Lemma 7.3 (2), we have for every ǫ> 0 ,

p p ui(x) u f(x) p dv = lim | | dv = | | dv i p p M Bǫ(x ) | | M Bǫ(x ) r M Bǫ(x ) r (7.38) Z \ 0 →∞ Z \ 0 Z \ 0 u p = χM Bǫ(x ) | | dv, \ 0 rp ZM where χM Bǫ(x0) is the characteritic function on M Bǫ(x0). As ǫ 0 , monotone convergence\ theorem and (7.38) imply that \ → p p ui u (7.39) f(x) p dv = lim | | dv = | | dv. | | i rp rp ZM →∞ ZM ZM Substituting (7.39) into (7.37) we obtain the desired (7.27) for u W 1,p(M).  ∈ 0 Remark 7.2. Theorem 7.7. is sharp and recaptures a theorem of Chen-Li-Wei (see [12]), when A =1 . A double limiting argument and techniques in [42] are employed in proving Theorem 7.7.

8. Monotonicity Formulae 2 Let F : [0, ) [0, ) be a C function such that F ′ > 0 on [0, ) , and F (0) = 0. ∞ → ∞ ∞

Definition 8.1. The F -degree dF is defined to be

tF ′(t) (8.1) dF = sup . t 0 F (t) ≥ Let ω be a smooth k-forms on a smooth n-dimensional Riemannian manifold M with values in the vector bundle ξ : E M. At each fiber of E is equipped with → 2 a positive inner product , E . Set ω = ω,ω E . The F,g-energy functional given by h i | | h i E ω 2 (8.2) F,g(ω)= F (| | )dvg . E 2 ZM The stress-energy tensor SF,ω associated with the F,g-energy functional is de- fined as follows (see [6, 5, 4, 28, 17]): E ω 2 ω 2 (8.3) SF,ω(X, Y )= F (| | )g(X, Y ) F ′(| | ) iX ω,iY ω , 2 − 2 h i where iX ω is the interior multiplication by the vector field X. 52 S.W. WEI

k Definition 8.2. ω A (ξ) (k 1) is said to satisfy an F -conservation law if SF,ω ∈ ≥ is divergence free, i.e. the (0, 1)-type tensor field div SF,ω vanishes identically

(8.4) div SF,ω 0. ≡ Let ♭ denote the bundle isomorphism that identifies the vector field X with the differential one-form X♭, and let be the Riemannian connection of M. Then the covariant derivative X♭ of X♭ is∇ a (0, 2)-type tensor, given by ∇ ♭ ♭ (8.5) X (Y,Z)= Z X Y = Z X, Y , X, Y Γ(M) . ∇ ∇ h∇ i ∀ ∈ If X is conservative, then

(8.6) X = f, X♭ = df and X♭ = Hess(f) . ∇ ∇ for some scalar potential f (see [11], p. 1527). A direct computation yields (see, e.g., [17])

♭ (8.7) div(iX Sω)= SF,ω, X + (div SF,ω)(X) , X Γ(M) . h ∇ i ∀ ∈ It follows from the divergence theorem that, if ω satisfies an F -conservation law, we have for every bounded domain D in M with C1 boundary ∂D ,

♭ (8.8) SF,ω(X,ν)dsg = SF,ω, X dvg , h ∇ i Z∂D ZD where ν is unit outward normal vector field along ∂D with (n 1)-dimensional 1−2 volume element dsg. When we choose scalar potential f(x) = 2 r (x), curvature via (8.6) and our Hessian comparison theorems will influence the behavior of the F - stress energy SF,ω and the underlying criticality ω with the help from the following.

Lemma 8.1. ([17]) Let M be a complete manifold with a pole x0. Assume that there exist two positive functions h1(r) and h2(r) such that

(8.9) h (r)[g dr dr] Hess(r) h (r)[g dr dr] 1 − ⊗ ≤ ≤ 2 − ⊗ on M x . If h (r) satisfies \{ 0} 2 (8.10) rh (r) 1 , 2 ≥ then

2 ♭ ω (8.11) SF,ω, X 1 + (n 1)rh (r) 2kdF rh (r) F (| | ) , h ∇ i ≥ − 1 − 2 2
where X = r r. ∇ Theorem 8.1. Let (M,g) be an n-dimensional complete Riemannian manifold with k a pole x0. Let ξ : E M be a Riemannian vector bundle on M and ω A (ξ). Assume that the radial→ curvature K(r) of M satisfies one of the following∈ seven conditions: DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS53

(8.12) (i) (3.14) holds with 1 + (n 1)A 2kdF A> 0; − 1 − 1+ √1+4A1 (ii) (3.17) holds with 1 + (n 1) kdF (1 + √1+4A) > 0; − 2 − 1 1 (iii) (3.49) holds with 1 + (n 1)( B + ) kdF 1+ 1+4B (1 B ) > 0; − | − 2| 2 − 1 − 1 1+ √1 4B p
(iv) (3.45) holds with 1 + (n 1) − kdF (1 + 1+4B ) > 0; − 2 − 1 2 2 (v) α K(r) β with α> 0 ,β > 0 and (np 1)β 2kαdF 0; − ≤ ≤− − − ≥ (vi) K(r) = 0 with n 2kdF > 0; − A B (vii) K(r) with ǫ> 0 , A 0 , 0

B A n (n 1) 2ke 2ǫ dF > 0. − − 2ǫ − If ω satisfies an F -conservation law, then 1 ω 2 1 ω 2 (8.13) λ F (| | )dv λ F (| | )dv ρ B (x ) 2 ≤ ρ B (x ) 2 1 Z ρ1 0 2 Z ρ2 0 for any 0 <ρ1 ρ2, where (8.14) ≤ 1 + (n 1)A 2kdF A, if K(r) satisfies (i), − 1 − 1+√1+4A1 √ 1 + (n 1) 2 kdF (1 + 1+4A), if K(r) satisfies (ii),  − 1 1 −  1 + (n 1)( B + ) kdF 1+ 1+4B1(1 B1) , if K(r) satisfies (iii),  − | − 2 | 2 − − λ =  1+√1 4B  1 + (n 1) 2− kdF (1 + √1+4B1), if K(r) satisfies (iv),  − −α p
 n 2k dF , if K(r) satisfies (v),  − β n 2kdF , if K(r) satisfies (vi),  − A  B 2ǫ  n (n 1) 2ǫ 2ke dF , if K(r) satisfies (vii) .  − − −   1 2 Proof. Choose a smooth conservative vector field X = ( 2 r ) on M. If K(r) satisfies (8.12), then by Corollary 3.1, Theorem A, Corollary∇ 3.5, and Theorem 3.5, (8.10) holds. Hence Lemma 8.1 is applicable and by (8.11) , we have on Bρ(x ) x , for every ρ> 0, 0 \{ 0} 2 ♭ ω (8.15) SF,ω, X λF (| | ) , h ∇ i≥ 2

♭ ω 2 where λ is as in (8.14) . Thus, by the continuity of SF,ω, X and F ( | | ), and h ∇ i 2 (8.3) , we have for every ρ> 0, (8.15) holds on Bρ(x0) and ω 2 ∂ (8.16) ρ F (| | ) SF,ω(X, ) on ∂Bρ(x ) . 2 ≥ ∂r 0 It follows from (8.8), (8.15) and (8.16) that ω 2 ω 2 (8.17) ρ F (| | )ds λ F (| | )dv . 2 ≥ 2 Z∂Bρ(x0) ZBρ(x0) 54 S.W. WEI

Hence we get from (8.17) the following ω 2 F ( | | )ds ∂Bρ(x0) 2 λ (8.18) ω 2 . R F ( | | )dv ≥ ρ Bρ(x0) 2 The coarea formula implies thatR d ω 2 ω 2 F (| | )dv = F (| | )ds . dρ 2 2 ZBρ(x0) Z∂Bρ(x0) Thus, we have d ω 2 F ( | | )dv dρ Bρ(x0) 2 λ (8.19) ω 2 R F ( | | )dv ≥ ρ Bρ(x0) 2 for a.e. ρ> 0 . By integrationR (8.19) over [ρ1,ρ2], we have 2 2 ω ω λ λ ln F (| | )dv ln F (| | )dv ln ρ2 ln ρ1 . B (x ) 2 − B (x ) 2 ≥ − Z ρ2 0 Z ρ1 0 This proves (8.13). 

9. Vanishing Theorems In this section we list some results that are immediate applications of the mono- tonicity formulae in the last section. 9.1. Vanishing theorems for vector bundle valued k-forms. Theorem 9.1. Suppose the radial curvature K(r) of M satisfies the condition (8.12). If ω Ak(ξ) satisfies an F -conservation law (8.4) and ∈ ω 2 (9.1) F (| | ) dv = o(ρλ) as ρ 2 → ∞ ZBρ(x0) ω 2 where λ is given by (8.14) depending on the curvature K(r), then F ( | | ) 0 , and 2 ≡ hence ω 0. In particular, if ω has finite F,g-energy, then ω 0. ≡ E ≡ Proof. By Theorem 8.1, the monotonicity formula (8.13) holds. Let λ2 in ω 2 → ∞ (8.13). Then (9.1) implies that F ( | | ) 0 , and hence ω 0.  2 ≡ ≡ 9.2. Applications in F -Yang-Mills fields. Let R∇ be an F -Yang-Mills field, associated with an F -Yang-Mills connection on the adjoint bundle Ad(P ) of a ∇ principle G-bundle over a manifold M. Then R∇ can be viewed as a 2-form with values in the adjoint bundle over M, and by [17, Theorem 3.1], ω = R∇ satisfies an F -conservation law. We have the following vanishing theorem for F -Yang-Mills fields: Theorem 9.2. Suppose the radial curvature K(r) of M satisfies the condition (8.12) , in which k =2 . Assume F -Yang-Mills field R∇ satisfies the following growth condition 2 R∇ (9.2) F (| | ) dv = o(ρλ) as ρ , 2 → ∞ ZBρ(x0) where λ satisfies the condition (8.14) for k =2 . Then R∇ 0 on M. In particular, ≡ every F -Yang-Mills field R∇ with finite F -Yang-Mills energy vanishes on M. DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS55

Proof. This follows at once from Theorem 9.1 in which k = 2 and ω = R∇ . 

This theorem becomes the following vanishing theorem for p-Yang-Mills fields, p p 1 2 when F (t)= p (2t) ,p> 1 (Hence dF = 2 by Definition 8.1) : Theorem 9.3. Suppose the radial curvature K(r) of M satisfies the one of the following seven conditions:

(9.3) (i) (3.14) holds with 1 + (n 1)A 2pA > 0; − 1 − 1+ √1+4A (ii) (3.17) holds with 1 + (n 1) 1 p(1 + √1+4A) > 0; − 2 − 1 1 (iii) (3.49) holds with 1 + (n 1)( B + ) p 1+ 1+4B (1 B ) > 0; − | − 2| 2 − 1 − 1 1+ √1 4B p
(iv) (3.45) holds with 1 + (n 1) − p(1 + 1+4B ) > 0 ; − 2 − 1 (v) α2 K(r) β2 with α> 0,β > 0 andp (n 1)β 2pα 0; − ≤ ≤− − − ≥ (vi) K(r) = 0 with n 2p> 0; − A B (vii) K(r) with ǫ> 0 , A 0 , 0

B A n (n 1) 2pe 2ǫ > 0. − − 2ǫ −

Then every p-Yang-Mills field R∇ with the following growth condition vanishes:

1 p λ (9.4) R∇ dv = o(ρ ) as ρ p | | → ∞ ZBρ(x0) where (9.5) 1 + (n 1)A 2pA, if K(r) satisfies (i), − 1 − 1 + (n 1) 1+√1+4A1 p(1 + √1+4A), if K(r) satisfies (ii),  − 2 − 1 + (n 1)( B 1 + 1 ) p 1+ 1+4B (1 B ) , if K(r) satisfies (iii),  − | − 2 | 2 − 1 − 1  1+√1 4B(1 B) λ =  − −  1 + (n 1) 2 p p√1+4B1,
if K(r) satisfies (iv),  − α − −  n 2p , if K(r) satisfies (v), − β n 2p, if K(r) satisfies (vi),  − B A  n (n 1) 2pe 2ǫ , if K(r) satisfies (vii) .  2ǫ  − − −  In particular, every p-Yang-Mills field R∇ with finite p-energy vanishes on M. YM Corollary 9.1. Let M,N,K(r) , λ , and the growth condition (9.4) be as in Theorem 9.3, in which p =2 . Then every Yang-Mills field R∇ 0 on M. ≡

10. Liouville Type Theorems 10.1. Liouville type theorems for F -harmonic maps. Let u : M N be an F -harmonic map. Then its differential du can be viewed as a 1-form→ with values 1 in the induced bundle u− TN. Since ω = du satisfies an F -conservation law (8.4) , we obtain the following Liouville-type 56 S.W. WEI

Theorem 10.1. Let N be a Riemannian manifold. Suppose the radial curvature K(r) of M satisfies one of the seven conditions in (12.3). Then every F -harmonic map u : M N with the following growth condition is a constant: → du 2 (10.1) F (| | ) dv = o(ρλ) as ρ , 2 → ∞ ZBρ(x0) where λ is as in (12.4). In particular, every F -harmonic map u : M N with finite F -energy is a constant. → Proof. This follows at once from Theorem 9.1, in which k = 1 and ω = du .  Analogously, we have the following Liouville Theorem for p-harmonic maps: Theorem 10.2. Let N be a Riemannian manifold. Suppose the radial curvature K(r) of M satisfies one of the following seven conditions: (10.2) (i) (3.14) holds with 1 + (n 1)A pA > 0; − 1 − 1+ √1+4A 1+ √1+4A (ii) (3.17) holds with 1 + (n 1) 1 p > 0; − 2 − 2 1 1 1+ 1+4B (1 B ) (iii) (3.49) holds with 1 + (n 1)( B + ) p 1 − 1 > 0; − | − 2| 2 − p 2 1+ √1 4B 1+ √1+4B (iv) (3.45) holds with 1 + (n 1) − p 1 > 0; − 2 − 2 (v) α2 K(r) β2 with α> 0,β > 0 and (n 1)β pα 0; − ≤ ≤− − − ≥ (vi) K(r) = 0 with n pk > 0; − A B (vii) K(r) with ǫ> 0 , A 0 , 0

B A n (n 1) pe 2ǫ > 0. − − 2ǫ − Then every p-harmonic map u : M N with the following p-energy growth condition is a constant: → 1 (10.3) du p dv = o(ρλ) as ρ , p | | → ∞ ZBρ(x0) where (10.4) 1 + (n 1)A pA, if K(r) satisfies (i), − 1 − 1 + (n 1) 1+√1+4A1 p 1+√1+4A , if K(r) satisfies (ii),  − 2 − 2 1+√1+4B1(1 B1)  1 + (n 1)( B 1 + 1 ) p − , if K(r) satisfies (iii),  − | − 2 | 2 − 2 λ =  1+√1 4B 1+√1+4B1  1 + (n 1) 2− p 2 , if K(r) satisfies (iv) ,  − α −  n p , if K(r) satisfies (v), − β n p, if K(r) satisfies (vi),  − B A  2ǫ  n (n 1) 2ǫ pe , if K(r) satisfies (vii) .  − − − In particular, every p-harmonic map u : M N with finite p energy is a  constant. → − p 1 2 Proof. This follows immediately from Theorem 10.1 in which F (t) = p (2t) and p  dF = 2 . DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS57

Corollary 10.1. Let M,N,K(r) , λ and the growth condition (10.3) be as in Theorem 10.2, in which p = 2 . Then every harmonic map u : M N is a constant. →

11. Generalized Yang-Mills-Born-Infeld fields (with the plus sign) on Manifolds In [34], Sibner-Sibner-Yang consider a variational problem which is a generaliza- tion of the (scalar valued) Born-Infeld model and at the same time a quasilinear generalization of the Yang-Mills theory. This motivates the study of Yang-Mills- Born-Infeld fields on R4, and they prove that a generalized self-dual equation whose solutions are Yang-Mills-Born-Infeld fields has no finite-energy solution except the trivial solution on R4. In [17], Dong and Wei introduced the following notions: Definition 11.1. The generalized Yang-Mills-Born-Infeld energy functional with the plus sign on a manifold M is the mapping + : R+ given by YMBI C → + 2 (11.1) BI ( )= 1+ R∇ 1 dv . YM ∇ M || || − Z q The generalized Yang-Mills-Born-Infeld energy functional with the negative sign on + a manifold M is the mapping − : R given by YMBI C →

− ( )= (1 1 R 2)dv . YMBI ∇ − − || ∇|| ZM q + The associate curvature form R∇ of a critical connection of BI (resp. BI− ) is called a generalized Yang-Mills-Born-Infeld field with∇ theYM plus sign (resp.YMwith the minus sign) on a manifold. Theorem 11.1 (See [17]). Every generalized Yang-Mills-Born-Infeld field (with the plus sign or with the minus sign) on a manifold satisfies an F -conservation law. Theorem 11.2. Let the radial curvature K(r) of M satisfy one of seven conditions in (8.12) in which k = 2 and dF = 1 . Let R∇ be a generalized Yang-Mills-Born- Infeld field with the plus sign on M. If R∇ satisfies the following growth condition

2 λ 1+ R∇ 1 dv = o(ρ ) as ρ , B (x ) || || − → ∞ Z ρ 0 q where

(11.2) 1 + (n 1)A 4A, if K(r) satisfies (i), − 1 − 1 + (n 1) 1+√1+4A1 2√1+4A), if K(r) satisfies (ii),  − 2 − 1 + (n 1)( B 1 + 1 ) 2 1+ 1+4B (1 B ) , if K(r) satisfies (iii),  2 2 1 1  − | − 1+| √1 4−B − λ =  1 + (n 1) − 2(1 + √1+4B ), if K(r) satisfies (iv),  2 p 1
 − n 4−α , if K(r)satisfies (v),  − β n 4, if K(r) satisfies (vi),  − B A  n (n 1) 4e 2ǫ , if K(r) satisfies (vii),  − − 2ǫ −  + then its curvature R∇ 0. In particular, if R∇ has finite -energy, then  ≡ YMBI R∇ 0. ≡ Proof. By applying Theorem 11.1 and F (t)= √1+2t 1 to Theorem 8.1 in which 2 − dF =1 , and k =2 , for R∇ A (AdP ), the result follows immediately.  ∈ 58 S.W. WEI

12. Dirichlet Boundary Value Problems

We recall F -lower degree lF is defined to be

tF ′(t) (12.1) lF = inf . t 0 F (t) ≥

1 A bounded domain D M with C boundary is called starlike ( relative to x0 ) if there exists an inner point⊂ x D such that 0 ∈ ∂ (12.2) ,ν ∂D 0 , h∂rx0 i| ≥

∂ where ν is the unit outer normal to ∂D , and for any x D x0 ∂D , ∂r (x) is ∈ \{ } ∪ x0 the unit vector field tangent to the unique geodesic emanating from x0 to x. It is obvious that any disc or convex domain is starlike.

1 Theorem 12.1. Let D be a bounded starlike domain (relative to x0) with C bound- ary in a complete Riemannian n-manifold M. Let ξ : E M be a Riemannian vector bundle on M and ω A1(ξ) . Assume that the radial→ curvature K(r) of M satisfies one of the following∈ seven conditions: (12.3) (i) (3.14) holds with 1 + (n 1)A 2dF A> 0; − 1 − 1+ √1+4A1 (ii) (3.17) holds with 1 + (n 1) dF (1 + √1+4A) > 0; − 2 − 1 1 (iii) (3.49) holds with 1 + (n 1) B + dF 1+ 1+4B (1 B ) > 0; − | − 2| 2 − 1 − 1 1+ √1 4B
p
(iv) (3.45) holds with 1 + (n 1) − dF (1 + 1+4B ) > 0; − 2 − 1 2 2 (v) α K(r) β with α> 0,β > 0 and (np 1)β 2αdF 0; − ≤ ≤− − − ≥ (vi) K(r) = 0 with n 2dF > 0; − A B (vii) K(r) with ǫ> 0 , A 0 , 0

B A n (n 1) 2e 2ǫ dF > 0. − − 2ǫ −

1 1 Assume that lF 2 . If ω A (ξ) satisfies F -conservation law and annihilates any tangent vector η≥of ∂D, then∈ ω vanishes on D.

Proof. By the assumption, there exists a point x D such that the distance 0 ∈ function rx0 relative to x0 satisfies (12.2). Take X = r r, where r = rx0 . From the proof of Theorem 8.1, we know that ∇

2 ♭ ω (8.15) SF,ω, X λF (| | ) h ∇ i≥ 2 DUALITIES IN COMPARISON THEORYEMS AND BUNDLE-VALUED GENERALIZED HARMONIC FORMS59 in D, where λ is a positive constant given by (12.4) 1 + (n 1)A 2dF A, if K(r) satisfies (i), − 1 − 1+√1+4A1 √ 1 + (n 1) 2 dF (1 + 1+4A), if K(r) satisfies (ii),  − 1 1 −  1 + (n 1)( B + ) dF 1+ 1+4B1(1 B1) , if K(r) satisfies (iii),  − | − 2 | 2 − − λ =  1+√1 4B  1 + (n 1) 2− dF (1 + √1+4B1), if K(r) satisfies (iv),  − −α p
 n 2 dF , if K(r) satisfies (v),  − β n 2dF , if K(r) satisfies (vi),  − B A  2ǫ  n (n 1) 2ǫ 2e dF , if K(r) satisfies (vii) .  − − −  Sinceω A1(ξ) annihilates any tangent vector η of ∂D, we easily derive via (8.3) and (12.1),∈ the following inequality on ∂D ∂ S (X,ν)= rS ( ,ν) F,ω F,ω ∂r ω 2 ∂ ω 2 ∂ = r F (| | ) ,ν F ′(| | ) ω( ),ω(ν) 2 h∂r i− 2 h ∂r i (12.5)   ∂ ω 2 ω 2 ω 2 = r ,ν F (| | ) 2F ′(| | )| | h∂r i 2 − 2 2   ∂ ω 2 r ,ν F (| | )(1 2lF ) 0 . ≤ h∂r i 2 − ≤ From (8.8), (12.3) and (12.5), we have ω 2 0 λF (| | )dv 0 , ≤ 2 ≤ ZD which implies that ω 0.  ≡ Theorem 12.2. (Dirichlet problems for F -harmonic maps) Let M, D, and ξ be as in Theorem 12.1. Assume that the radial curvature K(r) of M satisfies one of the following seven conditions: (12.6) (i) (3.14) holds with 1 + (n 1)A 2dF A> 0 ; − 1 − 1+ √1+4A1 (ii) (3.17) holds with 1 + (n 1) dF (1 + √1+4A) > 0; − 2 − 2 2 (iii) α K(r) β with α> 0 ,β > 0 and (n 1)β 2dF α 0; − ≤ ≤− − − ≥ (iv) K(r) = 0 with n 2dF > 0; − A B (v) K(r) with ǫ> 0 , A 0 , 0

B A n (n 1) 2e 2ǫ dF > 0 ; − − 2ǫ − 1 1 (vi) (3.49) holds with 1 + (n 1)( B + ) dF 1+ 1+4B (1 B ) > 0 ; − | − 2| 2 − 1 − 1 1+ √1 4B p
(vii) (3.45) holds with 1 + (n 1) − dF (1 + 1+4B ) > 0 . − 2 − 1 1 p Let u : D N be an F -harmonic map with lF into an arbitrary Riemannian → ≥ 2 manifold N. If u ∂D is constant, then u D is constant. | | 60 S.W. WEI

Proof. Take ω = du. Then ω ∂D = 0. Hence ω satisfies an F -conservation law and annihilates any tangent vector| η of ∂D . The assertion follows at once from Theorem 12.1 and [17, Theorem 6.1]. 

Corollary 12.1. Suppose M and D satisfy the same assumptions of Theorem 12.2. Let u : D N be a p-harmonic map (p 1) into an arbitrary Riemannian manifold → ≥ N. If u ∂D is constant, then u D is constant. | | p p 1 2 Proof. For a p-harmonic map u, we have F (t) = p (2t) . Obviously dF = lF = 2 . Take ω = du. This corollary follows immediately from Theorem 12.1 or Theorem 12.2. 

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Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019- 0315, U.S.A. Email address: [email protected]