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∗ 1 ∗ Γ (f)+ ℓΓv (f) (Lf) + (ρ ℓ−1)Γ(f) + (ρ + ℓρ )Γv (f), 2 2 ≥ n 1 − 2,0 2,1 ∞ for any f C (M) and ℓ> 0. Here, Γ(f) and Γ2(f) is defined as in (1.1) and (1.2) ∈ ∗ ∗ with L = ∆′ , while Γv (f) = v∗(df, df) for some v∗ Γ(Sym2 TM) and Γv (f) h ∈ 2 is defined analogously to Γ2(f). We also gave a geometrical interpretation of these constants. A short summary of the results of Part I is given in Section 2. In this paper, we want to explore how this inequality can be used to obtain ′ results for the heat semigroup of ∆h. In Section 3, we will address the question of whether a smooth bounded function with bounded gradient under the action of the heat semigroup will continue to have a uniformly bounded gradient. This will be an important condition for the results to follow. For a complete Riemannian man- ifold, a sufficient condition for this to hold is that the Ricci curvature is bounded from below, see e.g. [22] and [19, Eq 1.4]. We are not able to give such a sim- ple formulation for the sub-Laplacian, however, we are able to prove that it holds in many cases, including fiber bundles with compact fibers and totally geodesic fibers. This was only previously only known to hold for sub-Riemannian manifolds with transverse symmetries of Yang-Mills type [8, Theorem 4.3], along with some isolated examples in [21, Section 4] and [10, Appendix]. We give several results using the curvature-dimension inequality of Part I that only rely on the bounded- ness of the gradient under the heat flow. Our results generalize theorems found in ′ [8, 6, 7]. In particular, if ∆h is a sub-Laplacian on (M, , h) satisfying our gener- alized curvature-dimension inequality, then under certainH conditions (analogous to positive Ricci curvature in Riemannian geometry) we have the following version of the Poincar´einequality 1 f f 2 df 2 ∗ . k − M kL (M,vol) ≤ √α k kL (h ) ∗ Here, α is a positive constant, h is the co-metric of ( , h), fM is the mean value of a compactly supported function f and for any η Γ(HT ∗M) we use ∈ ∗ η 2 ∗ := h (η, η) dvol . k kL (h ) ZM In Section 4 we look at results which require the additional assumption that ∗ ∗ Γv (f, Γ(f)) = Γ(f, Γv (f)). This is important for inequalities involving logarithms. CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 3
We give a description of what this condition means geometrically and discuss re- sults that follow from it, such as a Li-Yau type inequality and parabolic Harnack inequality. In Section 5 we give some concrete examples, mostly focused on case of sub- Riemannian structures appearing from totally geodesic foliations with a complete metric. Here, all previously mentioned assumptions are satisfied. In this case, we also give a comment on how the invariants in our sub-Riemannian curvature- dimension inequality compare to the Riemannian curvature of an extended metric. In parallel with the development of our paper, part of the results of Theorem 3.4 and Lemma 4.1 was given in [9] for the case of sub-Riemannian obtained from Riemannian foliations with totally geodesic leaves that are of Yang-Mills type.
1.1. Notations and conventions. Unless otherwise stated, all manifolds are con- nected. If M is any vector bundle over a manifold M, its space of smooth E → sections is written Γ( ). If s Γ( ), we generally prefer to write s x rather than s(x) for its value in xE M. By∈ aE metric tensor s on , we mean smooth| section ∈ E of Sym2 ∗ which is positive definite or at least positive semi-definite. For every E such metric tensor, we write e s = s(e,e) for any e even if s is only posi- tive semi-definite. All metrick tensorsk are denoted by bold,∈ E lower case Latin letters p (e.g. h, g,... ). We will only use the term Riemannian metric for a positive definite metric tensor on the tangent bundle. If g is a Riemannian metric, we will use g∗, k g∗,... for the metric tensors induced on T ∗M, k T ∗M,... . ∧ If α is a form on a manifold M, its contraction or interior product by a vector V field A will be denoted by either ιAα or α(A, ). We use A for the Lie derivative with respect to A. If M is furnished with a RiemannianL metric g, any bilinear tensor s : TM TM R can be identified with an endomorphism of TM using g. We use the notation⊗ tr→s( , ) for the trace of this corresponding endomorphism, with the metric being implicit.× × If is a subbundle of TM, we will also use the H notation trH s( , ) := tr s(prH , prH ), where prH is the orthogonal projection to . × × × × H 2. Summary of Part I In this section, we briefly recall the most important definitions and results from Part I.
2.1. Sub-Riemannian manifolds. A sub-Riemannian manifold is a triple (M, , h) where M is a connected manifold and h is a positive definite metric tensor definedH only on the subbundle of TM. Equivalently, it can be considered as a manifold with a positive semi-definiteH co-metric h∗ that is degenerate along a subbundle of T ∗M. This latter mentioned subbundle will be Ann( ), the annihilator of , that ∗ H H consist of all covectors vanishing on . Define ♯h : p h∗(p, ) TM. We will assume that the subbundle isH bracket-generating,7→ i.e. its sections∈H⊆ and their iterated brackets span the entireH tangent bundle. Then we have a well defined met- ric dcc on M by taking the infimum over the length of curves that are tangent to . H 2.2. Two notions of sub-Laplacian. Let vol be any smooth volume form on M. We then define the sub-Laplacian relative to the volume form vol as ∆hf = ∗ div ♯h df, where the divergence is defined relative to vol . From the definition, 4 E. GRONG, A. THALMAIER it is clear that ∆h is symmetric relative the measure vol, i.e. M f∆hg dvol = g∆ f dvol for any f,g C∞(M) of compact support. M h c R We also introduced the∈ concept of a sub-Laplacian defined relative to a comple- R ment of . Let g be any Riemannian metric satisfying g H = h and let be the orthogonalV H complement of . Consider the following connection,| V H (2.1) ˚ Z = pr pr Z + pr pr Z ∇A H ∇prH A H V ∇prV A V + prH[prV A, prH Z]+prV [prH A, prV Z], where is the Levi-Civita connection of g. We define the sub-Laplacian of as ∇ V ′ 2 ∆ = trH ˚ f. h ∇×,× It is simple to verify that this definition is independent of g V , it only depends on h and the splitting TM = . | H⊕V Remark 2.1. If is the vertical bundle of a submersion π : M B into a Riemann- ian manifold (B,V gq) and if h is a sub-Riemannian metric defined→ by pulling back gq to an Ehresmann connection on π, then the sub-Laplacian ∆′ of satisfies H h V ′ q ∆ (f π) = (∆f) π, h ◦ ◦ q where ∆ is the Laplacian of gq and f C∞(B). ∈ 2.3. Metric-preserving complement. A subbundle is integrable if [Γ( ), Γ( )] Γ( ). By the Frobenius Theorem, such a subbundle givesV us a foliation onVM. WeV ⊆ sayV that an integrable complement of is metric-preserving if V H pr∗ h =0, for any V Γ( ), LV H ∈ V where prH is the projection corresponding to the choice of complement . Let g ⊥ V be any Riemannian metric such that g H = h and = . If we define ˚ as in | H V ∇ (2.1), then is metric preserving if and only if ˚h∗ = 0. The foliation of is then called a RiemannianV foliation. ∇ V 2.4. Generalized curvature-dimension inequality. For a given smooth sec- ond order differential operator L without constant term and for any section s∗ of Sym2 TM, define ∗ ∗ ∗ Γs (f,g)= s∗(df, dg), Γs (f,f)= Γs (f), ∗ 1 ∗ ∗ ∗ ∗ Γs (f,g)= LΓs (f,g) Γ(Lf,g) Γs (f,Lg) , Γs (f,f)= Γs (f). 2 2 − − 2 2 Assume that 1 (L(fg) fLg gLf)= h∗(df, dg). 2 − − for some positive semi-definite section h∗ of Sym2 TM. We say that L satisfies the generalized curvature-dimension inequality (CD*) if there is another positive semi- definite section v∗ of Sym2 TM, a positive number 0 < n and real numbers ρ ,ρ and ρ such that for any ℓ> 0 and f C∞(M), ≤ ∞ 1 2,0 2,1 ∈ ∗ ∗ 1 ∗ ∗ (CD*) Γh +ℓv (f) (Lf)2 + (ρ ℓ−1)Γh (f) + (ρ + ℓρ )Γv (f). 2 ≥ n 1 − 2,0 2,1 Let (M, , h) be a sub-Riemannian manifold with being a bracket-generating subbundleH of TM. Assume that we have an integrableH metric-preserving comple- ment and let g be a Riemannian metric such that and are orthogonal, with V H V CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 5
∗ ∗ h = g H and v := g V . Let h and v be their respective co-metrics. Relative to these structures,| we make| the following assumptions. (i) We define the curvature of relative to the complement as the vector valued 2-form H V (A, Z)=pr [pr A, pr Z], A,Z Γ(TM). R V H H ∈ We assume that there is a finite, minimal positive constant M < such R ∞ that (v, ) g∗ ⊗ g M pr v g for any v TM. Since M is never kR k ≤ Rk H k ∈ R zero when = 0, we can normalize v by requiring M = 1. Let m be the V 6 R R ∗ maximal constant satisfying α( ( , )) ∧2h∗ mR α v pointwise for any ∗ k R k ≥ k k α Γ(T M). Note that mR can only be non-zero if is bracket-generating of∈ step 2, i.e. if and its first order brackets span theH entire tangent bundle. H ∇˚ (ii) Define RicH(Z1,Z2) = tr A R (prH A, Z1)Z2 . This is a symmetric 2- tensor, which vanishes for vectors7→ in . We assume that there is a lower V bound ρ for Ric , i.e. for every v TM, we have H H ∈ Ric (v, v) ρ pr v 2 . H ≥ Hk H kh ∗ (iii) Write M ∗ = sup ˚ v (, ) ∗ ∗ and assume that it is finite. ∇˚v M ∇ g ⊗ Sym2 g Define
′ ∗ 2 ∗ (∆ v )(α, α) =tr H(˚ v )(α, α) h ∇×,× ′ ∗ 2 ∗ and assume that (∆ v )(α, α) ρ ′ v∗ α ∗ pointwise for any α Γ(T M). h ≥ ∆h k kv ∈ (iv) Finally, introduce RicHV as 1 RicHV (A, Z)= tr g(A, (˚ )( ,Z)) + g(Z, (˚ )( , A)) . 2 ∇×R × ∇×R × Assume then that RicHV (Z,Z) 2MHV pr Z v pr Z h pointwise. ≥− k V k k H k These assumptions guarantee that the sub-Laplacian ∆′ of satisfies (CD*). h V s∗ ′ ′ Theorem 2.2. Define Γ2 with respect to L = ∆h. Then ∆h satisfies (CD*) with n = rank , H −1 ρ1 = ρ c , H − (2.2) 1 2 M M 2 ρ2,0 = mR c( HV + ∇˚v∗ ) , 2 − 1 ′ 2 ρ = ρ v∗ M ∗ , 2,1 2 ∆h − ∇˚v for any positive c> 0. See Part I, Section 3.2 for a geometric interpretation of these constants.
2.5. The case when ˚ preserves the metric. Assume that we can find a metric ∇ ˚ ∗ ′ tensor v on satisfying v = 0. Then ∆h = ∆h, where ∆h is defined relative to the volumeV form of the∇ Riemannian metric g defined by g∗ = h∗ + v∗. Hence, L = ∆h is symmetric with respect to this volume form and satisfies the inequality
∗ ∗ 1 ∗ ∗ (CD) Γh +ℓv (f) (Lf)2 + ρ ℓ−1 Γh (f)+ ρ Γv (f) 2 ≥ n 1 − 2 6 E. GRONG, A. THALMAIER with ρ = ρ c−1, 1 H − (2.3) 1 ρ = m2 cM 2 , 2 2 R − HV for any positive c> 0. We shall also need the following result. Proposition 2.3. For any f C∞(M), and any c> 0 and ℓ> 0, ∈ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ Γh (Γh (f)) Γh (f) Γh +ℓv (f) (̺ ℓ−1)Γh (f) ̺ Γv (f) , 4 ≤ 2 − 1 − − 2 1 ∗ ∗ ∗ ∗ Γh (Γv (f)) Γv (f)Γv (f), 4 ≤ 2 where ̺ = ρ c−1 and ̺ = cM 2 . 1 H − 2 − HV 2.6. Spectral Gap. Let (M, , h) be a compact sub-Riemannian manifold where is bracket-generating. Let LHbe a smooth second order operator without constant H ∗ term satisfying qL = h and assume also that L is symmetric with respect to some volume form vol on M. Assume that L satisfies (CD*) with ρ2,0 > 0. Let λ be any nonzero eigenvalue of L. Then nρ k 2,0 ρ 2 λ, k = max 0, ρ . n + ρ (n 1) 1 − ρ ≤− 2 { − 2,1} 2,0 − 2,0 3. Results under conditions of a uniformly bounded gradient 3.1. Diffusions of second order operators. Let T 2M denote the bundle of second order tangent vectors. Let L be a section of T 2M, i.e. a smooth second order differential operator on L without constant term. Consider the short exact sequence inc q 0 TM T 2M Sym2 TM 0 → −→ −→ → where qL = q(L) is defined by 1 (3.1) q (df, dg)= (L(fg) fLg gLf) , f,g C∞(M). L 2 − − ∈ Assume that qL is positive semi-definite. Then for any point x M and relative F P 1 ∈ to some filtered probability space (Ω, , ), we have a 2 L-diffusion X = X(x) defined up to some explosion time τ = τ(x), see [12, Theorems 1.3.4 and 1.3.6]. In other words, there exist an F -adapted M-valued semimartingale X(x) satisfying X (x)= x and such that for any f C∞(M), 0 ∈ 1 d(f(X )) Lf(X ) dt t − 2 t is the differential of a local martingale up to τ(x). The diffusion X(x) is defined on the stochastic interval [0, τ(x)), with τ(x) being an explosion time in the sense that the event τ(x) < is almost surely contained in limt↑τ Xt(x)= . For { ∞} ′ { ∞} a construction of Xt(x) in the case of L = ∆h, see Part I, Section 2.5. Let Pt be the corresponding semigroup Ptf(x) = E[1t≤τ f(Xt(x))] for bounded measurable functions f. Note that in general Pt1 1 with equality if and only if ≤ ∞ τ(x) = a.s. Also note that for any compactly supported f Cc (M), we have ∞1 ∈ 1 ∂tPtf = 2 LPtf. If τ = a.s., then ut = Ptf is the unique solution to ∂tut = 2 Lut with initial condition u ∞= f, where (t, x) u (x) is a smooth function on R M. 0 7→ t + × CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 7
Since qL is positive semi-definite, we can write L (non-uniquely) as
k 2 L = Zi + Z0, i=1 X where k is an integer and Z0,Z1,...,Zk are vector fields, not necessarily linearly independent at every point. If we assume that these vector fields and their brackets span the entire tangent bundle, then L is a hypoelliptic operator [11]. Hence, it has a smooth heat kernel with respect to any volume form on M. By [18], we also ∞ have Ptf > 0 for any nonnegative function f C (M), not identically zero, see also [13, Introduction]. We will only consider∈ such second order operators in this paper. ∗ ∗ Write h = qL. Assume that L satisfies (CD*) for some v . We want to use this inequality to obtain statements of Pt. However, we are going to need the following condition to hold to make such statements.
3.2. Boundedness of the gradient under the action of the heat semigroup. The most important property which we are going to need for all of our results, is the ∞ following condition. Let Cb (M) be the collection of all bounded smooth functions. ∞ h∗+v∗ ∞ We have Pt1= 1 and for any f Cb (M) with Γ (f) Cb (M) (A) ∈ h∗+v∗ ∈ and any T > 0, it holds that sup Γ (P f) ∞ < . t∈[0,T ] k t kL ∞ To understand condition (A) better, let us first discuss the special case when h = g is a Riemannian metric, v∗ = 0 and L = ∆ is the Laplacian of g. Then (CD*) holds if and only if the Ricci curvature is bounded from below, see e.g. [20]. If we in addition know that g is complete, then (A) is satisfied. However, even if we know that Pt1 = 1 and that the manifold is flat, condition (A) still may not hold if g is an incomplete metric. See [19] for a counter-example. We list some cases where we are ensured that (A) is satisfied. We expect there to be more cases where this condition holds.
3.2.1. Fiber bundles with compact fibers. Let L be a second order operator on a ∗ ∗ ∗ ∗ manifold M with qL = h . Let v be any other co-metric such that h + v is positive definite. The following observation was given in [21, Lemma 2.1, Proof (i)]. Lemma 3.1. Assume that there exists a function F C∞(M) and a constant C > 0 satisfying ∈ x : F (x) s is compact for any s> 0, •{LF CF≤, } ∗ ∗ • Γh +≤v (F ) CF 2. • ≤ Then (A) holds for the semigroup Pt of the diffusion of L.
Let (B, gq) be a complete n-dimensional Riemannian manifold with distance dgq and Ricci bound from below by ρ 0. For a given point b B, define r = dgq (b , ). ≤ 0 ∈ 0 Then the function F = √1+ r2 (or rather an appropriately smooth approximation) q satisfies the above conditions relative to ∆. This follows from the fact that (outside the cut-locus) Γgq (r) = 1 and from the Laplacian comparison theorem q 1 ∆r (n 1) + √ ρ . ≤ − r − 8 E. GRONG, A. THALMAIER
Now let π : M B be a fiber bundle with a compact fiber over this Rie- mannian manifold→B. Choosing an Ehresmann connection on π, we define a ∗ H sub-Riemannian manifold (M, , h) by h = π gq H. Then F π clearly satisfies H ′ ′ | ◦ Lemma 3.1 with respect to L = ∆h +Z where ∆h is the sub-Laplacian of = ker π∗ and Z is any vector field with values in . It follows that (A) holds in thisV case. V Remark 3.2. Let π : M B be a surjective submersion into a Riemannian mani- folds (B, gq). Let be an→ Ehresmann connection on π and define a sub-Riemannian H ∗ structure ( , h) by h = π gq H. In this case, π is a distance-decreasing map from the H | metric space (M, dcc) to (M, dgq ), where the metrics dcc and dgq are defined relative to ( , h) and gq, respectively. This follows from the observation that for any hori- zontalH curve γ in M from the point x to the point y, the curve π γ will be a curve ◦ of equal length in B connecting π(x) with π(y), hence d (x, y) dgq (π(x), π(y)). cc ≥ In particular, if dcc is complete, so is dgq , and the converse also hold if π is a fiber bundle with compact fibers. Furthermore, if ∆h is the sub-Laplacian of = ker π∗ satisfying (CD*), then the Ricci curvature of B is bounded from below,V since, by Remark 2.1, if we insert a function f π, f C∞(B) into (CD*), we obtain the usual curvature-dimension inequality on◦ B, ∈ gq 1 q gq Γ (f) (∆)2 + ρ Γ (f). 2 ≥ n 1 A result in [4, Prop 6.2] tells us that ρ1 must be a lower Ricci bound for B. We summarize all the above comments in the following proposition. Proposition 3.3. Let (M, , h) be a complete sub-Riemannian manifold with an integrable metric preservingH complement . Let be the foliation induced by ′ V F V and let ∆h be the sub-Laplacian of . Assume that the leafs of are compact and that M/ gives us a well definedV smooth manifold. Finally assumeF that L = ∆′ + Z satisfiesF (CD*) with respect to some v∗ on . Then (A) also hold for the h V corresponding semigroup Pt of L. Notice that in this case, unlike what we will discuss next, there is no requirement on the number of brackets needed of vector fields in in order to span the entire tangent bundle. H 3.2.2. A sub-Laplacian on a totally geodesic Riemannian foliation. Assume that (M, g) is a complete Riemannian manifold with a foliation given by an integrable subbundle . Let be the orthogonal complement of Fand assume that is V H V H bracket-generating. Write h = g H. Define ˚ relative to the splitting TM = | ∇ H⊕V as in (2.1). Assume that ˚ g = 0, which is equivalent to stating that is a metric preserving complement of∇ (M, , h) and that is a totally geodesic foliation.V Note H F that since g is complete, so is (M, dcc), where dcc is defined relative to the sub- Riemannian metric h. For such sub-Riemannian manifolds, the we can deduce the following.
Theorem 3.4. Let ∆h be the sub-Laplacian of the volume form of g or equiva- lently . Assume that ∆h satisfies the assumptions of Theorem 2.2 with m > 0. V R M 2 ∞ Let k = max ρH, HV 0. Then, for any compactly supported f Cc (M), ℓ> 0 and t {−0, } ≥ ∈ ≥ Γv∗ (P f) P Γv∗ (f) , t ≤ t p p CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 9
∗ ∗ ∗ ∗ 2 Γh (P f)+ Γv (P f) ekt/2P Γh (f)+ Γv (f) + (ekt/2 1), t t ≤ t k − q q 2 kt/2 where we interpret k (e 1) as t when k =0. As a consequence (A) holds. 1 − In particular, any ∆h-diffusion X(x) with X (x)= x M has infinite lifetime. 2 0 ∈
We remind the reader that mR > 0 can only happen if TM is spanned by and first order brackets of its sections. The proof is similar to the proof given forH the special case of sub-Riemannian manifolds with transverse symmetries of Yang- Mills type given in [8, Section 3 & Theorem 4.3]. In our terminology, these are sub-Riemannian manifolds with a trivial, integrable, metric-preserving complement satisfying MHV = 0. The key factors that allow us to use a similar approach are V Proposition 2.3 and the relation [∆h, ∆]f = 0, where ∆ is the Laplace operator of g and f C∞(M). The latter results follow from Lemma A.1 (c) in the Appendix. Since the∈ proof uses spectral theory and calculus on graded forms, it is left to Appendix A.3.2. Theorem 3.4 also holds in some cases when is not an integrable subbundle. See Appendix A.5 for details. V
3.3. General formulation. Let L be an operator as in Section 3.1 with corre- 1 sponding 2 L-diffusion X(x) satisfying X0(x) = x and semigroup Pt. We will as- ∗ sume that L satisfies (CD*) with v and the constants n,ρ1,ρ2,0 and ρ2,1 being implicit. Note that if L satisfies (CD*) for some value of the previously mentioned constants, then L also satisfies the same inequality for any larger n or smaller values of ρ1,ρ2,0 or ρ2,1. For the remainder of the section, no result will depend on n, however, we will need condition (A) to hold. Our proofs rely on the fact that, for any smooth function (t, x) u (x) C∞([0, ) M, R), 7→ t ∈ ∞ × we have a stochastic process Yt = ut Xt such that dYt equals ((∂t + L)ut) Xt dt modulo differentials of local martingales.◦ Hence, if (∂ + L)u 0 and if◦ u () t t ≥ t is bounded for every fixed t, then Yt is a (true) submartingale and E[Yt] is an increasing function with respect to t. In our presentation, we will usually state the result for a smooth, bounded func- ∞ h∗+v∗ ∞ tion f Cb (M) with bounded gradient Γ (f) Cb (M). Our results gener- alize theorems∈ found in [8, 6, 7]. ∈ We will first construct a general type of inequality, from which many results can be obtained. See [21, Theorem 1.1 (1)] for a similar result, with somewhat different assumptions. Lemma 3.5. Assume that L satisfies the conditions (CD*) and (A). For any T > 0, let a,ℓ C([0,T ], R) be two continuous functions which are smooth and positive on (0,T∈). Assume that there exist a constant C, such that 1 a˙ (t) (3.2)a ˙(t)+ ρ a(t)+ C 0, ℓ˙(t)+ ρ + ρ + ℓ(t) 0, 1 − ℓ(t) ≥ 2,0 2,1 a(t) ≥ holds for every t (0,T ). Then ∈ ∗ ∗ ∗ ∗ a(0)Γh +ℓ(0)v (P f) a(T )P Γh +ℓ(T )v (f)+ C P f 2 (P f)2 T ≤ T T − T ∗ ∗ for any f C∞(M) with Γh +v (f) C∞(M). ∈ b ∈ b 10 E. GRONG, A. THALMAIER
Proof. Define ut by ut(x)= PT −tf(x) for any 0 t T , x M. For any x M, consider the stochastic process ≤ ≤ ∈ ∈ ∗ ∗ Y (x) := a(t)Γh +ℓ(t)v (u ) X (x)+ Cu2 X (x). t t ◦ t t ◦ t Write loc= for equivalence modulo differentials of local martingales. Then, if (3.2) holds ∗ ∗ ∗ ∗ dY loc= a˙(t)Γh +ℓ(t)v (u )+ a(t)ℓ˙(t)Γv (u )+ CΓh (u ) X dt t t t t ◦ t h∗+ℓ(t)v∗ + a(t)Γ2 (ut) Xtdt ◦ ∗ a˙ (t) + (ρ ℓ(t)−1)a(t)+ C Γh (u ) X dt ≥ 1 − t ◦ t a˙(t) ∗ + a(t) ℓ˙(t)+ + ρ + ρ ℓ(t) Γv (u ) X dt 0. a(t) 2,0 2,1 t ◦ t ≥ Since Yt is bounded by (A), it is a true submartingale. Hence h∗+ℓ(T )v∗ 2 E[YT ]= a(T )PT Γ (f)+ CPT f ∗ ∗ E[Y ]= a(0)Γh +ℓ(0)v (P f)+ C(P f)2. ≥ 0 T T
3.4. Gradient bounds. We give here the first results that follow from Lemma 3.5. Proposition 3.6. Assume that L satisfies conditions (CD*) and (A). Let f ∗ ∗ ∈ C∞(M) be any smooth bounded function satisfying Γh +v (f) C∞(M). b ∈ b (a) For any constant ℓ> 0, if α(ℓ) = min ρ 1 ,ρ + ρ2,0 , then 1 − ℓ 2,1 ℓ ∗ ∗ ∗ ∗ Γh +ℓv (P f) e−α(ℓ)tP Γh +ℓv (f). t ≤ t (b) Assume that ρ > 0 and let k = max 0, ρ and k = max 0, ρ . Then 2,0 1 { − 1} 2 { − 2,1} ∗ 2 k tΓh (P f) 1+ + k + 2 t (P f 2 (P f)2). t ≤ ρ 1 ρ t − t 2,0 2 (c) Assume that ρ 0, ρ 0 and ρ > 0. Then 1 ≥ 2,1 ≥ 2,0 −ρ t 1 e 1 ∗ 2 − Γh (P f) 1+ P f 2 (P f)2 , ρ t ≤ ρ t − t 1 2,0 where we interpret (1 e−ρ1t)/ρ as t when ρ =0. − 1 1 (d) Assume that ρ1, ρ2,0 and ρ2,1 are nonnegative. Then for any ℓ> 0
ℓ ∗ ∗ P f 2 (P f)2 tP Γh +(ℓ+t)v (f). ℓ + t t − t ≤ t Proof. For all of our results (a)–(c), we will use Lemma 3.5. (a) Let ℓ(t)= ℓ be a constant, choose C = 0 and put a(t)= e−α(ℓ)t. Then (3.2) is satisfied and we obtain ∗ ∗ ∗ ∗ Γh +ℓv (P f) e−α(ℓ)T P Γh +ℓv (P f). T ≤ T T (b) For any T 0, consider a(t)= T t and ℓ(t)= ρ2,0 (T t). Then ≥ − T k2+2 − a˙(t) ℓ˙(t)+ ρ + ρ + ℓ(t) 0, 2,0 2,1 a(t) ≥ CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 11
and −1 T k2 +2 a˙(t) + (ρ1 ℓ(t) )a(t) 1 k1 , − ≥− − − ρ2,0 so (3.2) is satisfied if we define C =1+ k T + T k2+2 . Using Lemma 3.5, we 1 ρ2,0 obtain ∗ ρ2,0 ∗ h + T v 2 2 T Γ T k2+2 (P f) C(P f ) C(P f) . T ≤ T − T (c) Since the case ρ1 = 0 is covered in (b), we can assume ρ1 > 0. Define 1 e−ρ1(T −t) a(t)= − ρ1 and let T −ρ1(T −t) t a(s) ds e 1+ ρ1(T t) ℓ(t)= ρ2,0 = ρ2,0 − − . a(t) ρ (1 e−ρ1(T −t)) R 1 − Note that limt↑T ℓ(t) = 0, while limt↑T a(t)/ℓ(t)=2/ρ2,0. The latter number is also an upper bound for a(t)/ℓ(t) since
T 2 d a(t) a(t) 2˙a(t) t a(s) ds + a(t) = > 0 dt ℓ(t) T 2 ρ2,0(R t a(s)ds) from the fact that R T 2˙a(t) a(s)ds + a(t)2 Zt 1 −ρ1(T −t) −ρ1(T −T ) −ρ1(T −t) 2 = 2 2e e 1+ ρ1(T t) + (1 e ) ρ1 − − − − 1 −ρ1(T −t) −2ρ1(T −t) = 2 2ρ1(T t)e +1 e . ρ1 − − − and that s 1 e−2s 2xe−2s is an increasing function, vanishing at s = 0. We can then7→ define− C =1+− 2 such that a(t),ℓ(t) and C satisfies (3.2). ρ2,0 (ℓ+T )t ℓ (d) Define a(t)= t, ℓ(t)= T and C = ℓ+T , then (3.2) is satisfied. −
We see here that the results of (a) and (d) cannot be stated independently of a choice of co-metric v∗. However, in the case of (a), this does help us to get global statements that are independent of v∗. 3.5. Bounds for the L2-norm of the gradient and the Poincar´einequality. We want to use an approach similar to what is used in [6, Corollary 2.4] to obtain a global inequality from the pointwise estimate in Proposition 3.6 (a) which is independent of v∗. Lemma 3.7. Let L Γ(T 2M) be a second order operator without constant term ∗ ∈ and with qL = h positive semi-definite. Assume also that there exists a volume form vol, such that
fLgdvol = gLf dvol, f,g C∞(M), ∈ c ZM ZM ∞ and that L is essentially self-adjoint on compactly supported functions Cc (M). 12 E. GRONG, A. THALMAIER
∞ Let Ptf be the semigroup defined as in Section 3.1 and let b : Cc (M) [0, ) R be any function such that × ∞ → h∗ ∞ (3.3) Γ (P f) 1 b(f,t), for any f C (M), t> 0. k t kL ≤ ∈ c t/T Assume that β(f,t) := limT →∞ b(f,T ) exist for every t> 0. Then h∗ h∗ ∞ Γ (P f) 1 β(f,t) Γ (f) for any f C (M). k t kL ≤ k k ∈ c Proof. Denote the unique self-adjoint extension of L an operator on L2(M, vol) by the same letter, and let Dom(L) be its domain. Then et/2Lf is the unique solution 2 1 ∞ in L (M, vol) of equation ∂tut = 2 Lut with initial condition u0 = f Cc (M). 2 2 ∈t/2L Since Ptf is in L (M, vol) whenever f is in L (M, vol), we have Ptf = e f (see Appendix A.3 for more details). ∗ Notice that since qL = h is positive semi-definite, the self-adjoint operator L is nonpositive. Let , denote the inner product on L2(M, vol). Consider the h i ∞ h∗ spectral decomposition L = λdE . Then since Γ (f) 1 = f,Lf , while − 0 λ k kL −h i h∗ Γ R(Ptf) 1 = f,LP2tf , k kL −h i the H¨older inequality tells us that for any 0
Proof. (a) By Proposition 3.6 (a), we have h∗ −α(ℓ)t h∗+ℓv∗ Γ (P f) 1 e Γ (f) 1 k t kL ≤ k kL ∞ with α(ℓ) = min ρ1 1/ℓ,ρ2,1 + ρ2,0/ℓ holds for any f Cc (M). It follows h∗ { − −α(ℓ)t h∗ } ∈ that Γ (Ptf) L1 e Γ (f) L1 from Lemma 3.7. For every t, we then take thek infimumk over≤ ℓ to getk k −αt −kt inf e e with k = min ρ1,ρ2,1 . ℓ ≤ { } (b) With α(ℓ) defined as in the proof of (a), note that if ρ1 ρ2,1 and if ρ2 > 1, then ≥ − ρ ρ + ρ inf e−α(ℓ)t = exp 2,0 1 2,1 t = e−αt ℓ − ρ +1 2,0 which gives us the first part of the result. For the second part, we assume that ρ1 >ρ2,1, since if α> 0 with ρ1 = ρ2,1, then we can always decrease ρ2,1 while keeping α positive. For two compactly supported functions f,g C∞(M), note that ∈ c t d (P f f)g dvol = P f g ds dvol t − ds s ZM ZM Z0 t t 1 1 ∗ = (∆ P f)g dvol ds = Γh (P f,g) dvol ds. 2 h s 2 s Z0 ZM Z0 ZM Hence, by the Cauchy-Schwartz inequality t 1 h∗ 1/2 h∗ 1/2 (P f f)g dvol Γ (P f) ∞ Γ (g) d vol, t − ≤ 2 k s kL ZM Z0 ZM
which has upper bound
1/2 t 1 ∗ ρ +1 ∗ ∗ Γh (f)+ 2,0 Γv (f) Γh (g)1/2 dvol e−αsds, 2 ρ ρ ∞ 1 2,1 L ZM Z0 − by Proposition 3.6 (a). From the spectral theorem, we know that Ptf reaches
an equilibrium P∞f which is in Dom(L) and satisfies LP∞f = 0. Since this h∗ implies Γ (P∞f) = 0, we must have that P∞f is a constant. ∞ Assume that vol(M)= . Then P∞f = 0 and hence, for any f,g Cc (M), we have ∞ ∈ 1/2 1 ∗ ρ +1 ∗ ∗ fg dvol Γh (f)+ 2,0 Γv (f) Γh (g)1/2 dvol . ≤ 2α ρ ρ ∞ ZM 1 − 2,1 L ZM ∞ However, since g is complete, we can find a sequence of functions fn Cc (M) g∗ ∈ such that fn 1 while Γ (fn) L∞ 0. Inserting such a sequence for f in the above formula↑ andk lettingkn → , we obtain the contradiction that ∞ → ∞ M g dvol = 0 for any g Cc (M). (c) Follows from the identity∈ R 2 2 2 1 f fM L2 = f dvol f dvol k − k M −vol(M) M Z ∞ Z ∂ 2 = (Ptf) dvol dt − 0 ∂t M Z∞ Z h∗ 1 h∗ = Γ (P f) dvol dt Γ (f) 1 . t ≤ αk kL Z0 ZM 14 E. GRONG, A. THALMAIER
4. Entropy and bounds on the heat kernel ∗ ∗ 4.1. Commutating condition on Γh and Γv . For some of our inequalities involving logarithms, we will need the following condition. Let L Γ(T 2M) be a ∈ ∗ second order operator without constant term with positive semi-definite qL = h defined as in (3.1). Assume that L satisfies either (CD*) or (CD) with respect to positive semi-definite v∗. We say that condition (B) holds if ∗ ∗ ∗ ∗ (B) Γh (f, Γv (f)) = Γv (f, Γh (f)) forevery f C∞(M). ∈ We make the following observation. Lemma 4.1. Let g be a Riemannian metric on a manifold M, with an orthogonal splitting TM = ⊥ and use this decomposition to define the connection ˚ H⊕ V ∗ ∗ ∇ as in (2.1). Write g H = h and g V = v and let h and v be their respective corresponding co-metrics.| Then | ∗ ∗ ∗ ∗ Γh (f, Γv (f)) = Γv (f, Γh (f)) holds for every f C∞(M) if and only if ˚v∗ = ˚h∗ =0. ∈ ∇ ∇ Proof. It is simple to verify that for any A Γ( ) and V Γ( ), we have ∈ H ∈ V ∗ ∗ ∇˚ ˚Ah =0, ˚V v =0,T (A, V )=0, ∇ ∇ ∗ ∗ where T ∇˚ is the torsion of ˚. Define ♯h as in Section 2 and let ♯v be defined ∇ analogously. Using the properties of ˚, we get ∇ h∗ v∗ v∗ h∗ h∗ 2 v∗ 2 Γ (f, Γ (f)) Γ (f, Γ (f)) = (♯ df) df ∗ (♯ df) df ∗ − k kv − k kh v∗ h∗ =2˚ h∗ df(♯ df) 2˚ v∗ df(♯ df) ∇♯ df − ∇♯ df ∗ ∗ + (˚ h∗ v )(df, df) (˚ v∗ h )(df, df) ∇♯ df − ∇♯ df ∗ ∗ = (˚ h∗ v )(df, df) (˚ v∗ h )(df, df). ∇♯ df − ∇♯ df Since T ∗M = ker h∗ ker v∗ and since ˚ preserves these kernels, the above expres- ⊕ ∇ sion can only vanish for all f C∞(M) if ˚h∗ = 0 and ˚v∗ = 0. ∈ ∇ ∇ Let L, Pt and X(x) be as in Section 3.1. In this section, we explore the results we obtain when both conditions (A) and (B) hold. We will also assume that L satisfies (CD) rather than (CD*). The reason for this is that in the concrete case when L is the sub-Laplacian of a sub-Riemannian manifold with an integrable metric-preserving complement, the condition (B) along with the assumptions of Theorem 2.2 imply (CD), see Section 2. For most of the results, we also need the requirement that ρ2 > 0. This means that we can use the results of [8, 6, 7]. Let us first establish some necessary identities. Let Pt be the minimal semigroup 1 ∗ ∞ of 2 L where qL = h . For a given T > 0, let ut := PT −tf with f C (M) ∞ 1 ∂ s∗ s∗ ∗ ∈ 2 ∩ L (M). It is clear that ( 2 L + ∂t )Γ (ut) = Γ2 (ut) for any s Γ(Sym TM). Also note that if F : U R R be a smooth function, then for any∈ f C∞(M) with values in U, we obtain⊆ → ∈ ∗ LF (f)= F ′(f)Lf + F ′′(f)Γh (f). Straight-forward calculations lead to the following identities. CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 15
Lemma 4.2.
(a) If ut = PT −tf has values in the domain of F , then
1 ∂ 1 ∗ L + F (u )= F ′′(u )Γh (u ). 2 ∂t t 2 t t In particular, if ut is positive then ∗ 1 ∂ Γh (u ) L + log u = t , 2 ∂t t − 2u2 t h∗ 1 ∂ Γ (u ) 1 ∗ L + u log u = t = u Γh (log u ). 2 ∂t t t 2u 2 t t t (b) For any s∗ Γ(Sym2 TM), we have ∈ 1 ∂ ∗ L + u Γs (log u ) 2 ∂t t t ∗ ∗ ∗ ∗ ∗ = u Γs (log u )+ u Γh (log u , Γs (log u )) Γs (log u , Γh (log u )) . t 2 t t t t − t t 1 ∂ h∗ h∗ ∗ In particular, L + utΓ (log ut)= utΓ2 (log ut). If v is any co-metric such ∗ ∗ 2 ∂t∗ ∗ ∗ ∗ that Γh (f, Γv (f)) = Γv (f, Γh (f)), then 1 L + ∂ u Γv (log u )= u Γv (log u ) 2 ∂t t t t 2 t as well. 4.2. Entropy bounds and Li-Yau type inequality. We follow the approach of [3], [8, Theorem 5.2] and [21, Theorem 1.1]. Lemma 4.3. Assume that L satisfies (CD). Also assume that (A) and (B) hold. Consider three continuous functions a,b,ℓ : [0,T ] R with a(t) and ℓ(t) being non-negative. Let C be a constant. Assume that a(t→),b(t) and ℓ(t) are smooth for t (0,T ) and on the same domain satisfy ∈ 0 a˙ (t)+ ρ 1 2b(t) a(t)+ C ≤ 1 − ℓ(t) − (4.1) a˙ (t) ( 0 ℓ˙(t)+ ρ2 + ℓ(t). ≤ a(t) ∞ h∗+v∗ Consider a positive function f Cb (M), f > 0 with bounded gradient Γ (f) ∞ ∈ ∈ Cb (M). Then we have
∗ ∗ ∗ ∗ a(0) P f Γh +ℓ(0)v (log P f) a(T )P fΓh +ℓ(T )v (log f) T T − T 2C (P (f log f) (P f) logP f) ≤ T − T T T T + n a(t)b(t)2dt P f 2 a(t)b(t)dt P Lf. T − T Z0 ! Z0 !
Proof. We have Ptf > 0 from our assumptions on L and f. For any T > 0, define u = P − f for 0 t T and t T t ≤ ≤ ∗ ∗ Y = a(t) u Γh (log u )+ ℓ(t)u Γv (log u ) X t t t t t ◦ t t +2C (u log u ) X + a(s) nb(s)2u 2b(s)Lu X ds. t t ◦ t s − s ◦ s Z0 16 E. GRONG, A. THALMAIER
Let us write loc= for equivalence modulo differentials of local martingales. We use that h∗ Γ (ut) h∗ Lut = utL log ut + = utL log ut + utΓ (log ut) ut and (CD) to obtain
∗ dY loc= (˙a(t) 2a(t)b(t)+ C) u Γh (log u ) X dt t − t t ◦ t ∗ + a˙(t)ℓ(t)+ a(t)ℓ˙(t) Γv (log u ) X dt t ◦ t h∗+ℓ(t)v∗ + a(t)u Γ (log u ) X dt t 2 t ◦ t + a(t)u nb(t)2 2b(t)L log u X dt t − t ◦ t 1 ∗ a˙(t)+ ρ 2b(t) a(t)+ C u Γh (log u ) X dt ≥ 1 − ℓ(t) − t t ◦ t a˙(t) ∗ + a(t) ℓ˙(t)+ ρ + ℓ(t) Γv (log u ) X dt 2 a(t) t ◦ t + na(t)u (b(t) L log u )2 X dt. t − t ◦ t Y is then a submartingale from (4.1). The result follows from E[YT ] E[Y0]. ≥
We look at some of the consequences of Lemma 4.3.
Corollary 4.4. Assume that L satisfies (CD) with ρ2 > 0, and that (A) and (B) ∞ h∗+v∗ also hold. Let f Cb (M) be any bounded smooth function with Γ (f) ∞ ∈ ∈ Cb (M). (a) (Entropy bound) Assume that ρ 0 and that f > 0. Then for any x M, 1 ≥ ∈ −ρ t 1 e 1 ∗ 2 f f − Γh (log P f)(x) 1+ P log (x). 2ρ t ≤ ρ t P f(x) P f(x) 1 2 t t (b) (Li-Yau inequality) Assume that n< in (CD) and that f 0, not identically zero. Then for any 1 <β< 2 and for∞ any t 0, ≥ ≥ ∗ Γh (P f) P Lf n a2 (4.2) t (a b ρ t) t β ρ t(2a b ρ t) (P f)2 − β − β 1 P f ≤ 4t (2 β)(β 1) − 1 β − β 1 t t − − ! ρ2 + β β 1 where aβ = and bβ = − . ρ2 β The special case of β = 2/3 in (4.2) is described with consequences in [8, The- orem 6.1]. If ρ 0, then for many application β = (2 + ρ )(1 + ρ ) ρ is a 1 ≥ 2 2 − 2 better choice, as this minimizes the ratio of a2 /(2 β)(β 1) over a . With this β − p − β choice, we obtain relation h∗ 1 Γ (Ptf) PtLf N (4.3) 2 , D (Ptf) − Ptf ≤ t where n (√2+ ρ + √1+ ρ )2 (2 + ρ )(1 + ρ ) (4.4) N := 2 2 ,D = 2 2 . 4 ρ2 p ρ2 CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 17
∞ Proof. Recall that if f Cb (M) is non-negative and non-zero, then Ptf is strictly positive. ∈ (a) We will use Lemma 4.3. As in Proposition 3.6 (c), for any T 0, define ≥ T −ρ1(T −t) −ρ1(T −t) 1 e t a(s) ds e 1+ ρ1(T t) a(t)= − , ℓ(t)= ρ2,0 = ρ2,0 − −ρ (T −t) − ρ1 a(t) ρ (1 e 1 ) R 1 − and C =1+2/ρ . If we define b(t) 0, condition (4.1) is satisfied. Hence, 2 ≡ −ρ T h∗+ ρ2T v∗ 1 e 1 Γ 2 (P f) 2 − T 1+ P (f log f) (P f) log P f . ρ P f ≤ ρ T − T T 1 T 2 Divide by PT f and evaluate at x for the result. (b) For any ε > 0, define fε = f + ε > 0. For any α > 0 and T > 0, define ℓ(t)= ρ2 (T t), a(t) = (T t)α+1 and α+2 − − 1 a˙ 1 1 α +2 1 b(t)= ρ + = ρ α +1+ . 2 1 a − ℓ 2 1 − ρ T t 2 − Note that T 1 ρ α +2 a(t)b(t) dt = 1 T α+2 1+ T α+1 , 2 α +2 − ρ (α + 1) Z0 2 T 1 ρ2 α +2 a(t)b(t)2 dt = 1 T α+2 2ρ 1+ T α+1 4 α +2 − 1 ρ (α + 1) Z0 2 (α + 1)2 α +2 2 + 1+ T α . α ρ (α + 1) 2 ! If we put C = 0, then (4.1) is satisfied and so if we use fε in Lemma 4.3 and let ε 0, we get ↓ h∗+ ρ2T v∗ Γ α+2 (P f) ρ α +2 T + 1 T 1+ P Lf P f α +2 − ρ (α + 1) T T 2 n ρ2 α +2 (α + 1)2 α +2 2 1 1 T 2ρ 1+ + 1+ P f. ≤ 4 α +2 − 1 ρ (α + 1) α ρ (α + 1) T T 2 2 ! Define β := (α + 2)/(α + 1) to obtain (4.2).
Using (4.3) and the approach found in [8, Remark 6.2 and Section 7] and [7], we obtain the following results. ∗ Corollary 4.5. Assume that L satisfies (CD) relative to v with ρ1 0,ρ2 > 0 and n < . Write g∗ = h∗ + v∗. Also assume that (A) and (B) hold≥ and that L ∞ is symmetric with respect to the volume form vol. Let pt(x, y) be the heat kernel of 1 2 L with respect to vol. Finally, let N and D be as in (4.4). Then the following holds. −N/2 (a) pt(x, x) t p1(x, x) for any x M. (b) For any≤0 ∗ where dcc is the Carnot-Carath´eodory distance. If g is the co-metric of a complete Riemannian metric, then t N/2 d (y,z)2 p (x, y) p (x, z) 1 exp D cc . t0 ≤ t1 t 2(t t ) 0 1 − 0 There are several more results which we can obtain when (A) and (B) hold, along with the fact that L satisfies (CD) with ρ2 > 0, which can be found in [8, 6, 7]. We list some of the most important results here, found in [8, Theorem 10.1] and [7, Theorem 1.5]. Theorem 4.6. Let L be a second order operator satisfying (CD) with respect to v∗ and with ρ2 > 0. Assume that it is symmetric with respect to some volume form vol. Define g∗ = h∗ + v∗ and assume that this is a complete Riemannian metric. Finally, assume that conditions (A) and (B) hold. Let Br(x) be the ball of radius r centered at x M with respect to the metric d . ∈ cc (a) (Sub-Riemannian Bonnet-Myers Theorem) If ρ1 > 0, then M is compact. (b) (Volume doubling property) If ρ 0, there exist a constant C such that 1 ≥ vol(B (x)) C vol(B (x)), for any r 0. 2r ≤ r ≥ (c) (Poincar´einequality on metric-balls) If ρ1 0, there exist a constant C such that ≥ ∗ f f 2 dvol Cr2 Γh (f)d vol, k − Br k ≤ ZBr (x) ZBr (x) for any r 0 and f C1 B¯ (x) where f = vol(B (x))−1 f dvol. ≥ ∈ r Br r ZBr (x) 5. Examples and comments 5.1. Results in the case of totally geodesic Riemannian foliations. Let us consider the following case. Let (M, g) be a Riemannian manifold, and let be a subbundle that is bracket generating of step 2, i.e. the tangent bundle is spannedH by the sections of and their first order brackets. Let be the orthogonal complement H V of with respect to g. Define ˚ with respect to the decomposition TM = andH let h and v be the respective∇ restrictions of g to and . Let us makeH⊕V the following assumptions: H V - is integrable, g is complete, ˚ g = 0 and the assumptions (i)–(iv) of Section 2 V ∇ hold with mR > 0. From our investigations so far, we then know that - is a metric-preserving complement of (M, , h); the foliation of is a totally geodesicV Riemannian foliation. H V - the sub-Laplacian ∆h of is symmetric with respect to the volume form vol of g; V ∞ - ∆h is essentially self-adjoint on Cc (M); ∗ - ∆h satisfies (CD) with respect to v ; - both (A) and (B) hold. We list the results that can be deduced on such manifolds using the approach of the generalized curvature-dimension inequality. We will split the results up into two propositions. CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 19 1 2 M 2 Proposition 5.1. Define κ = 2 mRρH HV and assume that κ 0. Let f ∞ − ≥ ∈ Cb (M) be non-negative, not identically zero. Define 2 2ρH + κ + ρH + κ n (κ + ρH)(κ +2ρH) N = q q ,D = . 4 κ q κ ∗ ∗ (a) Assume that Γh +v (f) C∞(M). Then for any 1 <β< 2, we have ∈ b ∗ ρ 2 Γh (P f) ρ P Lf n 1+ H β t 1+ H β t 2κ . (P f)2 − 2κ P f ≤ 4t (2 β)(β 1) t ! t − − 1 (b) Let pt(x, y) be the heat kernel of 2 ∆h with respect to vol. Then 1 p (x, x) p (x, x) t ≤ tN/2 1 for any x M and 0 t 1. Furthermore, for any 0 Proof. From the formulas (2.3), we know that ∆h satisfies (CD) with ρ2 > 0 and ρ 0. In particular, we can choose c = 1/ρ if ρ > 0 and if ρ = 0. This 1 ≥ H H ∞ H M choice gives us ρ1 = 0, while maximizing ρ2. Note that if ρH = 0, then HV must be 0 as well, since we have required κ 0. ≥ Example 5.2 (Free nilpotent Lie algebra of step 2). Let h be a vector space of dimension n with an inner product , and let k denote the vector space 2 h. Define a Lie algebra g as the vector spaceh i h k with Lie brackets determined by k being the center and for any A, B h, we have⊕ V ∈ [A, B]= A B k,. ∧ ∈ This is clearly a nilpotent Lie algebra of step 2 and dimension n(n + 1)/2. Let G be a simply connected nilpotent Lie group with Lie algebra g. Define a sub-Riemannian structure ( , h) by left translation of h and its inner product. Let H n 2 A1,...,An be a left invariant orthonormal basis of and define L = i=1 Ai . From Part I, Example 4.4, we know that L satisfies (CDH ) with respect to some v∗, 1 ∗ P n = rank h, ρ1 = 0 and ρ2 = 2(n−1) . This choice of v also gives us a complete Riemannian metric g satisfying ˚ g = 0 and with L being the sub-Laplacian of the volume form of g. We then obtain∇ that for any 0 1 2 M 2 Proposition 5.3. Define κ = 2 mRρH HV and assume that κ> 0. Then the following statements hold. − (a) M is compact. (b) If f C∞(M) is an arbitrary function and ∈ 2 2κ α := , M 2 HV + mR 2ρH +2κ we have q h∗ −αt h∗ 2 1 h∗ Γ (P f) 1 e Γ (f) 1 , and f f 2 Γ (f) dvol k t kL ≤ k kL k − M kL ≤ α ZM −1 where fM = vol(M) M f dvol. (c) Let f C∞(M) be an arbitrary function. Then ∈ R h∗ 2ρH 2 2 tΓ (Ptf) 1+ (Ptf (Ptf) ). ≤ κ ! − (d) Let f be a strictly positive smooth function. Then for any x M, ∈ ∗ 2ρ f f tΓh (log P f)(x) 2 1+ H P log (x). t ≤ κ t f(x) f(x) ! Proof. From the formulas (2.3), we know that ∆h satisfies (CD) with ρ2 > 0 and ρ1 > 0. (a) Follows directly from Theorem (4.6) (a). (b) We use Propositions 3.6 (a) and 3.8 (b). With our assumption of ˚v = 0, the ∇ formulas (2.2) show that we can choose ρ2,1 = 0 and both ρ1 and ρ2 strictly positive, since κ> 0. The result follows by maximizing ρ2ρ1 with respect to c. ρ2+1 (c) We use Proposition 3.6 (c) and using (2.3) with c =1/ρH. (d) Similar to the proof of (c), only using Corollary 4.4 (a) instead. Example 5.4. Let g be a compact semisimple Lie algebra with bi-invariant metric 1 A, B = tr ad(A) ad(B), ρ> 0. h i −4ρ Let G be a (compact) Lie group with Lie algebra G and with metric gq given by left (or right) translation of the above inner product. Then ρ> 0 is the lower Ricci bound of G. Let h be the subspace of the Lie algebra g g consisting of elements on the form (A, 2A), A g. Define the subbundle on⊕G G by left translation of h. If we use the same∈ symbol for an element inH the Lie× algebra and the corresponding left invariant vector field, we define a metric h on by H h((A, 2A), (A, 2A)) = A, A . h i Define π : G G G as projection on the second coordinate with vertical bundle × → = ker π∗ and give this bundle a metric v determined by V 1 (A, 0) 2 = A, A . k kv 4ρh i CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 21 If we then define ˚ relative to and g = pr∗ h + pr∗ v, then ˚ g = 0. Let ∇ H⊕V H V ∇ ∆h be the sub-Laplacian with respect to , which coincides with the sub-Laplacian of the volume form vol of g. We showedV in Part I, Example 4.6 that this satisfies ∗ 1 2 (CD) with respect to v , n = dim G, ρ1 = ρH =4ρ and ρ2 = 2 mR =1/4. We then have that for any f C∞(G G), ∈ × 2 5 h∗ f f × 2 = Γ (f) dvol k − G GkL 4ρ ZM −1 where f × = vol(G G) f dvol . G G × G×G 5.2. Comparison to RiemannianR Ricci curvature. Let us consider a sub- Riemannian manifold such as in Section 5.1. Given the results of Proposition 5.1 and Proposition 5.3, it seems reasonable to consider sub-Riemannian manifolds with κ 0 or κ > 0 as the analogue of Riemannian manifolds with respectively non-negative≥ and positive Ricci curvature. However, given the extra structure in the choice of v on , it is natural to ask how these sub-Riemannian results compare V ∗ ∗ to the Ricci curvature of the metric g = prH h + prV v. We give the comparison here. Introduce the following symmetric 2-tensor ˚ Ric (Y,Z)=tr V pr R∇(V, Y )Z . V 7→ V Then the Ricci curvature of g can be written in the following way. Proposition 5.5. The Ricci curvature Ricg of g satisfies 1 2 (5.1) Ricg(Y, Y ) = Ric (Y, Y ) + RicHV (Y, Y )+ g(Y, (, )) 2 ∗ H 2k R k∧ g 3 2 + Ric (Y, Y ) (Y, ) ∗ . V − 4kR kg ⊗ g Before we get to the proof, let us note the consequences of this result. If κ = 1 2 M 2 2 ρHmR HV is respectively non-negative or positive, this ensures that the first line of (5.1)− has respectively a non-negative or positive lower bound. Furthermore, note that this part is independent of any covariant derivative of vertical vector fields. of Proposition 5.5. Let be the Levi-Civita connection of g. Define a two tensor ∇ (A, Z)= Z ˚ Z. Then it is clear that B ∇A − ∇A ˚ R∇(A, Y )Z = R∇(A, Y )Z + (˚ )(Y,Z) (˚ )(A, Z) ∇AB − ∇Y B + ( (Y, A),Z)+ (A, (Y,Z)) ( (A, Y ),Z) (Y, (A, Z)). B B B B −B B −B B Furthermore, it is simple to verify that 1 1 1 (A, Z)= (A, Z) ♯ g(A, (Z, )) ♯ g(Z, (A, )). B 2R − 2 R − 2 R Let A1,...,An and V1,...,Vν be local orthonormal bases of respectively and . Then H V n ˚ g(A , R∇(A ,Z)Z R∇(A ,Z)Z) i i − i i=1 X n 3 2 1 2 = g(Z, (˚ )(A ,Z)) (Z, ) ∗ + g(Z, (, )) 2 ∗ . ∇Ai R i − 4kR kg ⊗ g 2k R k∧ g i=1 X 22 E. GRONG, A. THALMAIER Similarly, ν g(R∇(V ,Z)Z R∇˚(V ,Z)Z, V ) = 0. s=1 s − s s 5.3. GeneralizationsP to equiregular submanifolds of steps greater than two. Many of the results in Section 3 and Section 4 depend on the condition ρ2 > 0. A necessary condition for this to hold is that our sub-Riemannian manifold is bracket-generating of step 2. Let us note some of the difficulties in generalizing the approach of this paper to sub-Riemannian manifolds (M, , h) of higher steps. As usual, we require bracket-generating. Assume also, forH the sake of simplic- ity, that is equiregular,H i.e. there exists a flag of subbundles = 1 2 3 H r such that H H ⊆ H ⊆ H ⊆···⊆H k+1 = span Z , [A, Z] : Z Γ( k), A Γ( ) , x M. Hx |x |x ∈ H ∈ H ∈ Choose a metric tensor v on and let g = pr∗ h + pr∗ v be the corresponding V H V Riemannian metric. Let be the orthogonal complement of k in k+1. Let Vk H H prVk be the projection to k relative to the splitting 1 r−1. Define ∗ V H⊕V ⊕···⊕V vk = v Vk and let vk be the corresponding co-metric. We could attempt to con- | h∗ v∗ v∗ struct a curvature-dimension inequality with Γ (f), Γ 1 (f),..., Γ r−1 (f). How- h∗ v∗ ever, a condition similar to (B) could never hold in this case, i.e. Γ (f, Γ k (f)) = v∗ h∗ Γ k (f, Γ (f)) cannot hold for any k r 2. ≤ − To see this let α and β be forms that only are non-vanishing on respectively k h∗ v∗ v∗ h∗ V and for k r 2. Then Γ (f, Γ k (f)) = Γ k (f, Γ (f)) holds if and only if Vk+1 ≤ − ˚h∗ = 0 and ˚ v∗ = 0 for any A Γ( ). Hence we obtain ∇ ∇A k ∈ H ∗ v∗ 0 = (˚ v )(α, β)= β([A, ♯ k α]). ∇A k However, this is a contradiction, since by our construction, k+1 must be spanned by orthogonal projections of brackets on the form [A, Z], A V Γ( ), Z Γ( ). ∈ H ∈ Vk Appendix A. Graded analysis on forms A.1. Graded analysis on forms. Let (M, , h) be a sub-Riemannian manifold with an integrable complement , and let vHbe a chosen positive definite metric ∗ ∗V tensor on . Let g = prH h + prV v be the corresponding Riemannian metric. The subbundleV gives us a foliation of M, and corresponding to this foliation we have a grading onV forms, see e.g. [2, 1]. Let Ω(M) be the algebra of differential forms on M. Let Ann( ) and Ann( ) be the subbundles of T ∗M of elements vanishing on respectively H and . IfV either a or b is a negative integer, then η Ω(M) is a homogeneousH elementV of degree (a,b) if and only if η = 0. Otherwise,∈ for nonnegative integers a and b, η is a homogeneous element of degree (a,b), if it is a sum of elements which can be written as α β, α Γ( a Ann( )), β Γ( b Ann( )). ∧ ∈ V ∈ H Relative to this grading, we canV split the exterior differentialV d into graded com- ponents d = d1,0 + d0,1 + d2,−1. The same is true for its formal dual δ = δ−1,0 + δ0,−1 + δ−2,1, CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 23 i.e. the dual with respect to the inner product on forms of compact support α, β, defined by (A.1) α, β = α ⋆β, α,β of compact support, h i ∧ ZM where ⋆ is the Hodge star operator defined relative to g. Note that δ−a,−b is the formal dual of da,b from our assumptions that and are orthogonal. We will give formulas for each graded component. H V A.2. Metric-preserving complement and local representation. We will use ♭ : TM T ∗M for the map v g(v, ) with inverse ♯. Let ˚ be defined as in (2.1) → 7→ ∇ relative to g and the splitting TM = . If α is a one-form and A1,...,An and V ,...,V are respective local orthonormalH⊕V bases of and , then locally 1 ν H V n ν dα = ♭A ˚ α + ♭V ˚ α α , i ∧ ∇Ai s ∧ ∇Vs − ◦ R i=1 s=1 X X and hence each of the three terms are local representations of respectively d1,0α, d0,1α and d2,−1α. Local representations on forms of all orders follow. Assume now that ˚ g = 0, i.e. is a metric-preserving compliment of ( , h) ∇ V H with a metric tensor v satisfying ˚v∗ = 0. From the formula of d1,0η = n ♭A ∇ i=1 i ∧ ˚ η, we obtain δ−1,0η = n ι ˚ η for any form η. Let ∆ be the sub- Ai i=1 Ai Ai h P Laplacian∇ of or equivalently− vol. Let∇ ∆ be the Laplacian of g. Then it is clear that for any fV C∞(M), we haveP ∈ −1,0 1,0 ∆f = δdf, ∆hf = δ d f. − − Lemma A.1. For any form η Ω(M), we have ∈ (A.2) δ−1,0d0,1α = d0,1δ−1,0α, δ0,−1d1,0α = d1,0δ0,−1α. − − ∞ As a consequence, for any f C (M) we have ∆h∆f = ∆∆hf. ∈ The following result is helpful for our computation in Part I, Lemma 3.3 (b) and Corollary 3.11. Lemma A.2. (a) For any horizontal A Γ( ), a vertical V Γ( ) and arbitrary vector filed Z Γ(TM), we have ∈ H ∈ V ∈ ˚ g(R∇(A, V )Z, A)=0. (b) If ˚ g =0, then for every point x0, there exist local orthonormal bases A1,...,An and∇ V ,...V , defined in a neighborhood of x , such that for any Y Γ(TM), 1 ν 0 ∈ 1 ˚ A = ♯ g(Z, (A , )) , ˚ V =0. ∇Z i|x0 2 R i |x0 ∇Z s|x0 of Lemma A.1. It is sufficient to show one of the identities in (A.2), since δ−1,0d0,1 is the formal dual of δ0,−1d1,0. From Lemma A.2 (a), any A Γ( ) and V Γ( ) satisfy ∈ H ∈ V ι ˚ ˚ α = ι ˚ ˚ α + ι ˚ α. A∇V ∇A A∇A∇V A∇[V,A] From the definition of ˚, it follows that T ∇˚ (A, V ) = 0, where T ∇˚ is the torsion of ∇ ˚. For a given point x M, let A ,...,A and V ,...,V be as in Lemma A.2 (b). ∇ 0 ∈ 1 n 1 ν 24 E. GRONG, A. THALMAIER All terms below are evaluated at the point x0, giving us ν n d0,1δ−1,0α = ♭V ˚ ι ˚ α − s ∧ ∇Vs Ai ∇Ai s=1 i=1 Xν Xn ν n ˚ ˚ ˚ = ♭Vs ι∇˚ Ai α ♭Vs ιAi Vs Ai α − ∧ Vs Ai ∇ − ∧ ∇ ∇ s=1 i=1 s=1 i=1 X X X X 1 ν n = g(V , (A , A ))♭V ι ˚ α −2 s R i j s ∧ Aj ∇Ai s=1 i,j=1 X X ν n ν n ˚ ˚ ˚ ♭Vs ιAi Ai Vs α ♭Vs ιAi ∇˚ A −∇˚ V α − ∧ ∇ ∇ − ∧ ∇ Vs i Ai s s=1 i=1 s=1 i=1 νXnX X X = ι ♭V ˚ ˚ α Ai s ∧ ∇Ai ∇Vs s=1 i,j=1 X X ν n = ι ˚ ♭V ˚ α = δ−1,0d1,0α. Ai ∇Ai s ∧ ∇Vs − s=1 i,j=1 X X Next, we prove the identity [∆h, ∆]f = 0. If we consider the degree (1, 1)-part of d2 = 0, we get d0,1d1,0 + d1,0d0,1 =0. 0,1 0,1 The same relation will then hold for their formal duals. Since ∆f = ∆hf δ d f, 0,−1 0,1 0,−1 0,1 − it is sufficient to show that ∆hδ d f = δ d ∆hf. This gives us the result 0,−1 0,1 −1,0 1,0 0,−1 0,1 0,−1 0,1 −1,0 1,0 0,−1 0,1 ∆hδ d f = δ d δ d f = δ d δ d f = δ d ∆hf, − − since we have to do an even number of permutations. A.3. Spectral theory of the sub-Laplacian. Let L be a self-adjoint operator on 2 2 2 2 L (M, vol) with domain Dom(L). Define f Dom(L) = f L2 + Lf L2 . Write the ∞ k k k k k k spectral decomposition of L as L = −∞ λdEλ with respect to the corresponding projector valued spectral measure Eλ. For any Borel measurable function ϕ: R R R ∞ → , we write ϕ(L) for the operator ϕ(L) := −∞ ϕ(λ)dEλ which is self adjoint on its domain R ∞ Dom(ϕ(L)) = f L2(M, vol) : ϕ(λ)2d E f,f . ∈ h λ i Z−∞ In particular, if ϕ is bounded, ϕ(L) is defined on the entire of L2(M, vol). See [15, Ch VIII.3] for details. Let (M, , h) be a sub-Riemannian manifold with sub-Laplacian ∆h defined relative toH a volume form vol. Assume that is bracket-generating and that H (M, dcc) is complete metric space, where dcc is the Carnot-Carath´eodory metric of ( , h). Then H f∆hg dvol = g∆hf dvol and f∆hf dvol 0. ≤ ZM ZM Z From [17, Section 12], we have that ∆h is a an essentially self adjoint operator on ∞ Cc (M). We denote its unique self-adjoint extension by ∆h as well with domain 2 Dom(∆h) L (M, vol). ⊆ CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 25 tλ/2 j tλ/2 Since ∆h is non-positive and the maps λ e and λ λ e are bounded on ( , 0] for t > 0, j > 0, we have that f7→ et/2∆h f is a7→ map from L2(M, vol) −∞∞ j 7→ 1 into Dom(∆ ). Define P f as in Section 3.1 with respect to ∆h-diffusions ∩j=1 h t 2 for bounded measurable functions f. Then clearly P f ∞ f ∞ . Since ∆h k t kL ≤ k kL is symmetric with respect to vol and P 1 1, we obtain P f 1 f 1 as well. t ≤ k t kL ≤k kL The Riesz-Thorin theorem then ensures that Ptf Lp f Lp for any 1 p . 2 k k 2≤k k ≤ ≤ ∞ In particular, Ptf is in L (M, vol) whenever f is in L (M, vol). This implies that P f = et/2∆h f for any bounded f L2(M, vol) by the following result. t ∈ Lemma A.3 ([14, Prop], [8, Prop 4.1]). Let L be equal to the Laplacian ∆ or sub-Laplacian ∆h defined relative to a complete Riemannian or sub-Riemannian 2 metric, respectively. Let ut(x) be a solution in L (M, vol) of the heat equation (∂ L)u =0, u = f, t − t 0 2 for a function f L (M, vol). Then ut(x) is the unique solution of this equation in L2(M, vol). ∈ t/2∆h Hence, we will from now on just write Pt = e without much abuse of notation. A.3.1. Global bounds using spectral theory. We now introduce some additional as- sumptions. Assume that g is a complete Riemannian metric with volume form vol, ⊥ such that g H = h, = and g V = v. Let ∆ be the Laplace-Beltrami operator | H V | v∗ of g and write ∆f = ∆hf + ∆vf where ∆vf = div ♯ df. Since g is complete, ∆ ∞ is also essentially self-adjoint on Cc (M) by [16] and we will also denote its unique self-adjoint extension by the same symbol. Assume that ˚ g = 0 where ˚ is defined as in (2.1). Recall that ∆h and ∆ ∞∇ ∇ commute on Cc (M) by Lemma A.1. Lemma A.4. (a) The operators ∆h and ∆ spectrally commute, i.e. for any bounded Borel func- tion ϕ : R R and f L2(M, vol), → ∈ ϕ(∆h)ϕ(∆)f = ϕ(∆)ϕ(∆h)f. Also Dom(∆) Dom(∆h). ⊆ (b) Assume that ∆h satisfies the assumptions of Theorem 2.2 with mR > 0. Then there exist a constant C = C(ρ ,ρ ) such that for any f C∞(M) Dom(∆2 ), 1 2 ∈ ∩ h 2 2 2 2 C f 2 = C f 2 + ∆hf 2 , k kDom(∆h) k kL k kL is an upper bound for h∗ h∗ v∗ v∗ Γ (f) dvol, Γ2 (f) dvol, Γ (f) dvol and Γ2 (f) dvol . ZM ZM ZM ZM ∞ Proof. (a) Note first that for any f Cc (M), using Lemma A.1 and the inner product (A.1) ∈ 0,−1 0,1 −1,0 1,0 0,1 0,1 −1,0 1,0 ∆vf∆hf dvol = δ d f,δ d f = d f, d δ d f h i h i ZM = d0,1f,δ−1,0d0,1d1,0f = d1,0d0,1f, d1,0d0,1f 0. −h i h i≥ 26 E. GRONG, A. THALMAIER Hence 2 2 2 (∆hf) dvol ((∆h + ∆v)f) dvol = (∆f) dvol, ≤ ZM ZM ZM and hence ∆hf L2 ∆f L2 is true for any f Dom(∆). We conclude k k ≤ k k t/2∆ ∈ that Dom(∆) Dom(∆h). Define Qt = e . It follows that, for any f ⊆ 2 ∈ Dom(∆h), ut = ∆hQtf is an L (M, vol) solution of ∂ 1 ∆ u =0, u = ∆hf. ∂t − 2 t 0 In conclusion, by Lemma A.3 we obtain ∆hQtf = Qt∆hf. For any s > 0 and f L2(M, vol), we know that Q f Dom(∆) ∈ s ∈ ⊆ Dom(∆h), and since 1 (∂ ∆h)Q P f =0, t − 2 s t it again follows from Lemma A.3 that PtQsf = QsPtf for any s,t 0 and f L2(M, vol). It follows that the operators spectrally commute,≥ see [15, Chapter∈ VIII.5]. (b) From Theorem 2.2, we know that ∆h satisfies (CD) with ρ2 > 0 and an appro- priately chosen value of c. The proof is otherwise identical to [8, Lemma 3.4 & Prop 3.6] and is therefore omitted. A.3.2. Proof of Theorem 3.4. We are going to prove that (A) holds without using stochastic analysis. We therefore need the following lemma. Lemma A.5 ([8, Prop 4.2]). Assume that (M, g) is a complete Riemannian man- ∞ ifold. For any T > 0, let u, v C (M [0,T ]), (x, t) ut(x), (x, t) vt(x) be smooth functions satisfying the∈ following× conditions: 7→ 7→ 2 T 2 (i) For any t [0,T ], ut L (M, vol) and 0 ut L dt < . ∈ ∈ T h∗ 1/2 k k ∞ For some p (ii) 1 p , 0 Γ (ut) L Rdvol < . ≤ ≤ ∞ qk k T ∞ (iii) For any t [0,T ], v L (M, vol) and v q dt < for some 1 q . ∈ t ∈R 0 k tkL ∞ ≤ ≤ ∞ Then, if (L + ∂ )u v holds on M [0,T ], we have ∂t ≥ × R t P u u + P v dt. T T ≥ 0 t t Z0 t/2∆h ∞ Let Pt = e . For given compactly supported f Cc (M) and T > 0, define function ∈ 1/2 v∗ 2 (A.3) z = Γ (P − f)+ ε ε, t,ε T t − with ε> 0,t [0,T ]. Since P f Dom(∆2 ), Lemma A.4 (b) tells us that, ∈ t ∈ h v∗ zt,ε 2 Γ (PT −t) vol C PT −tf 2 < , k kL ≤ ≤ k kDom(∆h) ∞ ZM so that z L2(M, vol). By Proposition 2.3, t,ε ∈ h∗ v∗ h∗ Γ (Γ (PT −tf)) v∗ (A.4) Γ (zt,ε) Γ2 (PT −tf). ≤ 4zt,ε + ε ≤ CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 27 From Lemma A.4 (b) it follows that both (i) and (ii) of Lemma A.5 is satisfied. Hence, using that from Proposition 2.3 1 (A.5) ∂ ∆h z t − 2 t,ε 1 v∗ v∗ 1 h∗ v∗ = Γ (P − f)Γ (P − f) Γ (Γ (P − f)) 0, 2(z + ε)3 T t 2 T t − 4 T t ≥ t,ε v∗ 2 1/2 v∗ 2 1/2 we get PT zT,ε = PT (Γ (f)+ ε ) PT ε z0,ε = (Γ (PT f)+ ε ) ε. By letting ε tend to 0, we obtain − ≥ − (A.6) Γv∗ (P f) P Γv∗ (f). T ≤ T h∗ p 2 1/2 p M 2 Next, let yt,ε = Γ (PT −tf)+ ε ε, choose any α > max ρH, HV 0 and define − {− } ≥ −α/2(T−t) v∗ ut,ε = e yt,ε + ℓΓ (PT −t(f)) . Note first that ∂ 1 + ∆h u ∂t 2 t,ε −α/2(T −t) e h∗ v∗ 1 h∗ h∗ = Γ (P − f)+ ℓy Γ (P − f) Γ (Γ (P − f)) 2y +2ε 2 T t t,ε 2 T t − 4y2 T t t,ε t,ε −α/2(T −t) αe h∗ v∗ + Γ (PT −tf)+ ε + ℓyt,sΓ (PT −tf) . 2yt,ε +2ε We use Proposition 2.3 with ℓ replaced by ℓyt,ε to get 1 h∗ h∗ h∗ −1 −1 −1 h∗ 2 Γ (Γ (PT −tf)) Γ2 (f) (ρH c ℓ yt,ε )Γ (f) 4yt,ε ≤ − − − ∗ ∗ + ℓy Γv (f) cM 2 Γv (f). t,ε 2 − HV ∂ 1 As a result, for any c> 0, ∂t + 2 ∆h ut,ε has lower bound −α/2(T −t) ∗ ∗ e −1 −1 −1 h M 2 v (ρH c ℓ yt,ε )Γ (PT −tf) c HV Γ (PT −tf) 2yt,ε +2ε − − − −α/2(T −t) αe h∗ v∗ + Γ (PT −tf)+ ℓyt,εΓ (PT −tf) . 2yt,ε +2ε Since it is true for any value of c> 0, it remains true for c = ℓyt,ε, and hence ∂ 1 e−α/2(T −t) + ∆h u . ∂t 2 t,ε ≥− ℓ In a similar way as before, we can verify that the conditions of Lemma A.5 hold by using Lemma A.4. We can hence conclude that −αT/2 v∗ u0,ε = e (y0,ε + ℓΓ (PT f)) T e−α(T −t)/2 P u + P dt ≤ T T,ε t ℓ Z0 ∗ 2 P y + ℓΓv (f) + 1 e−αT/2 . ≤ T T,ε αℓ − 28 E. GRONG, A. THALMAIER αT/2 M Multiplying with e on both sides, letting ε 0 and α k := max ρH, R , we finally get that for any ℓ> 0, → → {− } ∗ ∗ (A.7) Γh∗ (P f)+ ℓΓv (P f) ekT/2P Γh∗ (f)+ ℓΓv (f) + ℓ−1F (T ), T T ≤ T k q q where 2 kt/2 k (e 1) if k> 0, Fk(t)= − (t if k =0. v∗ v∗ −1/2 Since this estimate holds pointwise, it holds for ℓ = PT Γ (f) Γ (PT f) or ∗ ∗ − ℓ = at points where P Γv (f) Γv (P f) = 0. The resulting inequality is ∞ T − T (A.8) Γh∗ (P f) ekT/2P Γh∗ (f) + (ekT/2 + F (T )) P Γv∗ (f) Γv∗ (P f). T ≤ T k T − t Weq will now show how thisq inequality implies (A). Sincep g is complete, there ∞ exist a sequence of compactly supported functions gn C (M) satisfying gn 1 h∗+v∗ ∈ ↑ pointwise and Γ (g ) ∞ 0. It follows from equation (A.6) and (A.8) that k n kL → h∗+v∗ lim Γ (Ptgn) L∞ 0 n→∞ k k → as well. Hence, since Ptgn Pt1 and dPtgn g∗ approach 0 uniformly, we have g∗ → k k that Γ (Pt1) = 0. It follows that Pt1 = 1. ∞ To finish the proof, consider a smooth function f C (M) with f L∞ < h∗+v∗ ∞∈ k k and Γ (f) L∞ < . Define fn = gnf C (M). Then PT fn PT f pointwise.∞ k It followsk that ∞ ∈ → b b (A.9) dPT f(γ ˙ (t)) dt = lim dPT fn (γ ˙ (t)) dt n→∞ Za Za for any smooth curve γ :[a,b] M. We want to use the dominated convergence theorem to show that the integral→ sign and limit on the right side of (A.9) can be interchanged. h∗ Without loss of generality, we may assume that Γ (gn) L∞ < 1 for any n. We then note that k k h∗+v∗ h∗ Γ (fn) f L∞ + Γ (f) =: K < . ∞ ≤k k ∞ ∞ L L q q This relation, combined with (A.6) and (A.8), gives us ∗ ∗ kT/2 Γh +v (P f ) ∞ 2e + F (T )+1 K. k T n kL ≤ k q v∗ Furthermore, the dominated convergence theorem tells us that both PT Γ (fm fn) h∗ − and limn→∞ PT Γ (fn fm) approach 0 pointwise as n,m . By inserting − h∗+v∗ → ∞ fn fm into (A.6) and (A.8), we see that Γ (PT fn) at any fixed point is a Cauchy− sequence and hence convergent. We conclude that b b dPT f(γ ˙ (t))dt = lim dPT fn (γ ˙ (t))dt. a a n→∞ Z Z It follows that dPtf limn→∞ dPtf vanishes outside a set of measure zero along any curve, so − h∗+v∗ h∗+v∗ Γ (PT f) L∞ = lim Γ (PT fn) L∞ < . k k n→∞ k k ∞ CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 29 In conclusion, we have proven that condition (A) holds. Without any loss of gen- erality we can put ℓ = 1, since we can obtain all the other inequalities by replacing f with ℓf. 1 Remark A.6. If we know that any 2 ∆h-diffusion starting at a point has infinite lifetime then using Lemma A.4, we can actually make a probabilistic proof. We outline the proof here. We will only prove the inequality (A.6) as the proof of (A.7) is similar. 1 We will again use zt,ε as in (A.3). Let X = X(x) be an 2 ∆h-diffusion with ε ε ε X0(x)= x M. We define Z by Zt = zt,ε Xt. Then Z is a local submartingale by (A.5). By∈ using the Burkholder-Davis-Gundy◦ inequality, there exist a constant B such that t ε ε 1 E sup Z BE Z + z (x)+ E (∂ ∆h)z X ds s ≤ h it 0,ε s − 2 s,ε ◦ s 0≤s≤t Z0 ∗ hp i where Zε = t Γh (z ) X ds is the quadratic variation of Zε. By the Cauchy- h it 0 s,ε ◦ s Schwartz inequality and the bound (A.4), we get the conclusion R t E ε v∗ Z t p2t(x, x) Γ2 (PT −sf) L1 dt < h i ≤ s 0 k k ∞ hp i Z which means that E sup Zε < . Hence, Zε is a true submartingale, 0≤s≤t∧τ s ∞ giving us (A.6). A.4. Interpretation of RicHV . Let be any integrable subbundle. Choose a subbundle such that TM = . AnyV such choice of correspond uniquely to a constant rankH endomorphismH⊕V pr = pr : TM TMH. This can be considered V →V⊆ as a splitting of the short exact sequence TM F TM/ . Let Ω(M) be the the exterior algebra ofV →M with→Z Z-gradingV of Section A.1. Choose nondegenerate metric tensors × v Γ(Sym2 ∗) and gq Γ(Sym2(TM/ )∗) ∈ V ∈ V on and TM/ . Since ν ∗ n(TM/ )∗ is canonically isomorphic to n+ν T ∗M, theV choices of vV and gq givesV us⊕ a volumeV form vol on M. We also have an energyV functionalV defined on projections to . RelativeV to pr, ∗q ∗ V define a Riemannian metric gpr = F g + pr v. We introduce a functional E on the space of projections pr by 2 E(pr) = pr ∧2 g∗ ⊗ g dvol kR k pr pr ZM where pr is the curvature of = kerpr. We can only be sure that the integral is finiteR if M is compact, so weH will assume this, and consider our calculations as purely formal when this is not the case. Let = pr be the restriction of the Levi-Civita connection of g to . Intro- ∇ ∇ pr V duce a exterior covariant derivative of d∇ on -valued forms in the usual way, i.e. V for any section V Γ( ), we have d∇V = V and if α is a -valued k-form, while µ is a form in the∈ usualV sense, then ∇ V k d∇(α µ) = (d∇α) µ + ( 1) α dµ. ∧ ∧ − ∧ 1,0 0,1 2,−1 We can split this operator into graded components d∇ = d∇ + d∇ + d∇ and do −1,0 0,−1 2,−1 the same with its formal dual δ∇ = δ∇ + δ∇ + δ∇ . 30 E. GRONG, A. THALMAIER Proposition A.7. The endomorphism pr is a critical value of E if and only if δ−1,0 =0. In particular, if g satisfies ∇ R (A.10) trV ( g)( , )=0, for any A Γ( ), LA × × ∈ H then pr is a critical value if and only if RicHV =0. Recall from Part I, Section 2.4 that condition (A.10) is equivalent to the leafs of the foliation of being minimal submanifolds. If is the vertical bundle of a submersion π : M F B, then we can identify TM/ Vwith π∗TB. In this case, a critical value of E →can be considered as an optimal wayV of choosing an Ehresmann connection on π. Proof. We write id := idT M for the identity on TM. Let pr be a projection to and α : TM be any -values one-from with ker α. Define a curve in theV →V V V ⊆ space projections prt = pr+tα. Then g (v, v) := g (v, v)= g (v, v)+2tv(αv, pr v)+ t2v(αv, αv). t prt pr Let be the curvature of pr . Then Rt t (A, Z)= (A, Z)+ tα[(id pr)A, (id pr)Z] Rt R − − t (pr[αA, (id pr)Z] + pr[(id pr)A, αZ]) + O(t2). − − − If t = prt , then ∇ ∇ t 1 1 v∗ V = V + td∇α(A, V ) t♯ g(d∇α(A, ), V ), and ∇A ∇A 2 − 2 1 1 ⊤ 2 d∇t pr = d∇ pr + t(d∇α) t(d∇α) + td∇α + O(t ), t 2 1,1 − 2 1,1 where (d∇α)1,1 is the (1,1)-graded component of d∇α and ⊤ v((d∇α)1,1(A, V1), V2)= v((d∇α)1,1(A, V1), V2). Since = d2,−1 pr , we get Rt − ∇ t d E(pr ) = ( 2 g∗ g )(d2,−1 pr, d1,0α) dvol dt t ∧ pr ⊗ pr ∇ ∇ t=0 ZM = (h∗ v)(δ−1,0 , α) dvol . − ⊗ ∇ R ZM Hence, pr is a critical value if and only if δ−1,0 = 0. ∇ R We give a local expression for this identity. Let A1,...,An be a local orthonormal basis of . Then H n δ−1,0 = (˚˚ )(A , )+ (N, ) ∇ R − ∇Ak R k R Xk=1 1 ˚˚ where N is defined by gpr(A, N)= 2 trV ( prH A g)( , ) and is the (0,0)-degree component of the Levi-Civita connection,− i.e.L × × ∇ ˚˚ Z = pr pr Z + pr pr Z. ∇A H ∇A H V ∇A V This coincides with RicHV when (A.10) holds. CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART II 31 A.5. If is not integrable. Let (M, g) be a complete Riemannian manifold and let beV a bracket-generating subbundle of TM with orthogonal complement . H V Define ˚ as in (2.1) with respect to g and the splitting TM = and assume ∇ H⊕V that ˚ g =0. Let ∆h be the sub-Laplacian defined relative to or equivalently to ∇ V the volume form of g. Then it may happen that (CD) holds for ∆h even without assuming that is integrable. More precisely, we will need the condition V (A.11) tr (v, (v, ))=0, v TM, R R ∈ (A, Z)=pr [pr A, pr Z], (A, Z)=pr [pr A, pr Z], A,Z Γ(TM). R V H H R H V V ∈ We refer to and as respectively the curvature and the co-curvature of . In Part I,R SectionR 3.8, we showed that Theorem 2.2 and Proposition 2.3H hold with the same definitions and with not integrable, as long as (A.11) also holds. The same is true for Theorem 3.4. WeV give some brief details regarding this. First of all, in Section A.1, the exterior derivative d now also has a part of degree ( 1, 2), determined by − d−1,2f =0, d−1,2α = α , f C∞(M), α Γ(T ∗M), − ◦ R ∈ ∈ and hence, the co-differential has a degree (1, 2)-part. However, these do not have any significance for our calculations. More− troubling is the fact that both ˚ Lemma A.2 (a) and the formula for Z Vs x0 in Lemma A.2 (b) are false when is not integrable. However, (A.11) ensures∇ that| V n ∇˚ g(R (Ai, V )Z, Ai)=0 i=1 X for any orthonormal basis A1,...,An of and vertical vector field V , which is all we need for the proof of Lemma A.1. FurthermoreH the same proof is still holds even ˚ 1 if now Z Vs x0 = 2 ♯ (Z, ) x0 in Lemma A.2 (b), as the extra terms cancel out. 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