PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 12, December 2010, Pages 4479–4486 S 0002-9939(2010)10514-2 Article electronically published on July 22, 2010

LOWER VOLUME ESTIMATES AND SOBOLEV INEQUALITIES

STEFANO PIGOLA AND GIONA VERONELLI

(Communicated by Michael Wolf)

Abstract. We consider complete manifolds with asymptotically non-negative curvature which enjoy a Euclidean-type Sobolev inequality and we get an ex- plicit lower control on the volume of geodesic balls. In case the amount of negative curvature is small and the Sobolev constant is almost optimal, we deduce that the manifold is diffeomorphic to Euclidean space. This extends previous results by M. Ledoux and C. Xia.

1. Introduction A Riemannian manifold (M,, ) of dimension dim M = m>p≥ 1issaidto support a Euclidean-type Sobolev inequality if there exists a constant CM > 0such ∈ ∞ that, for every u Cc (M), 1 1 p∗ p p∗ p (1) |u| dvol ≤ CM |∇u| dvol , M M where mp p∗ = m − p and dvol denotes the Riemannian measure of M. Clearly, (1) implies that there ∗ exists a continuous imbedding W 1,p (M) → Lp (M)andcanbeexpressedinthe equivalent form − C p ≤ inf |∇u|p dvol, M ∈ u Λ M where ∗ p∗ p p Λ= u ∈ L (M):|∇u|∈L and |u| dvolM =1 . M The validity of (1), as well as the best value of the Sobolev constant CM ,have intriguing and deep connections with the geometry of the underlying manifold, many of which are discussed in the excellent lecture notes [5]. See also [8] for a survey in the more abstract perspective of Markov diffusion processes, and [9] for the relevance of (1) in the Lp,q-cohomology theory. For instance, we note that a complete manifold with non-negative Ricci curvature (but, in fact, a certain amount of negative curvature is allowed) and supporting a Euclidean-type Sobolev inequality is necessarily connected at infinity. This fact can be proved using (non- linear) potential-theoretic arguments; see [10], [11].

Received by the editors March 12, 2010. 2010 Mathematics Subject Classification. Primary 53C21; Secondary 46E35.

c 2010 American Mathematical Society Reverts to public domain 28 years from publication 4479

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It is known (see e.g. Proposition 4.2 in [5]) that

(2) CM ≥ K(m, p), where K (m, p) is the best constant in the corresponding Sobolev inequality of Rm. It was discovered by M. Ledoux, [7], that for complete manifolds of non-negative Ricci curvature, the equality in (2) forces M to be isometric to Rm.Thisimportant rigidity result has been generalized by C. Xia, [13], by showing that, in case CM is sufficiently close to K (m, p), then M is diffeomorphic to Rm. The key ingredient in the Ledoux-Xia argument is a sharp lower estimate for the growth of geodesic balls which depends explicitly on the Sobolev constant. Actually, it is known, [2], that the validity of (1) implies that there exists a (small) constant γ = γ(m, p, CM ) > 0 (depending continuously on CM ) such that

(3) V (Bt) ≥ γV (Bt). However, to obtain the desired rigidity, one needs to get a sharp value for γ.In this paper, using a somewhat more geometric approach, we are able to extend this kind of estimate to manifolds with asymptotically non-negative curvature; see Theorem 1. As a consequence, using a theorem by S.-H. Zhu, [14], we will deduce rigidity even in this more general context. Notation. In what follows, having fixed a reference origin o ∈ M,wesetr (x)= distM (x, o), and we denote by Bt and ∂Bt the geodesic ball and sphere of radius t>0 centered at o. The corresponding balls and spheres in the m-dimensional Euclidean space are denoted by Bt and ∂Bt. Finally, the symbols V (Bt)and A (∂Bt) stand, respectively, for the Riemannian volume of Bt and the (m − 1)- dimensional Hausdorff measure of ∂Bt. Theorem 1. Let (M,, ) be a complete, m-dimensional Riemannian manifold, m ≥ 3,with (4) M Ric(x) ≥−(m − 1)G(r(x)) on M for some non-negative function G ∈ C0([0, +∞)). Assume that G satisfies the integrability condition ∞ (5) tG(t)dt = b<+∞ 0 and that the Euclidean-type Sobolev inequality (1) holds on M,forsome1

0

Corollary 2. Given m ≥ 3, m>p, there exist constants b0 = b0(m, p) > 0 and ε0 = ε0(m, p) > 0 such that the following holds:

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Let M be an m-dimensional complete manifold supporting the Sobolev inequality (1) with CM ≤ K(m, p)+ε,forsome0 <ε≤ ε0, and such that M Sect ≥−G(r(x)) on M, m where G satisfies (5) for some 0

2. Proof of the lower volume estimate m Recall that, in R , the equality in (1) with the best constant CRm = K(m, p) is realized by the (radial) Bliss-Aubin-Talenti functions φλ (x)=ϕλ(|x|) for every λ>0, where |x| is the Euclidean norm of x and ϕλ(t) are the real-valued functions defined as m−p β (m, p) λ p2 ϕλ (t)= m − . p p 1 λ + t p−1 If we choose β(m, p) > 0 such that p∗ (8) φλ (x)dx =1, Rm then −p |  | | |p K (m, p) = ϕλ ( x ) dx Rm

and, by the standard calculus of variations, the extremal functions φλ obey the (non-linear) Yamabe-type equation ∗ Rm − −p p −1 (9) Δpφλ = K (m, p) φλ , where p−2 (10) Δpu =div |∇u| ∇u stands for the p-Laplacian of a given function u. Define ˆ ˆ φλ : M → R as φλ(x):=ϕλ(r(x)). If we were in the assumptions of Ledoux and Xia, namely, if the Ricci curvature was non-negative, the idea of our proof would be simply to apply the Karp version p−2 of Stokes’ theorem, [6], to the vector field Xλ := φˆλ |∇φλ| ∇φˆλ,oncewehave ˆ observed that, by (9) and the Laplacian comparison theorem, each function φλ on M satisfies ∗ ˆ ≥− −p ˆp −1 Δpφλ K (m, p) φλ . This would lead directly to inequality (2.2) in [13], and the desired volume con- clusion can be deduced. The strategy for the proof of the general case stated in Theorem 1 is completely similar. Clearly, this time we have to take into account the (small) perturbations of (10) introduced by the negative curvature. Note also that, due to the possible presence of cut-points, all the computations have to be performed in the sense of distributions. Proof of Theorem 1. Let h ∈ C2([0, +∞)) be the solution of the problem h(t) − G(t)h(t)=0, h(0) = 0,h(0) = 1,

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m−1 and consider the m-dimensional model manifold Mh defined as Mh := (R × S , ds2 + h2(s)dθ2), where dθ2 is the standard metric on Sm−1. We shall use an index Bh h to denote objects and quantities referred to Mh. Thus, we denote by t and Bh ∂ t the geodesic ball and sphere of radius t>0inMh. Moreover, we introduce the family of functions φλ,h : Mh → R defined by φλ,h((s, θ)) := ϕλ(s). For later purposes, we recall that ([14], [10])

Bh ≥ B ≥ (11) V( t ) V( t), t 0.

Furthermore, we observe that, according to the Bishop-Gromov comparison theo- Bh rem and its generalizations, [3], [10], A (∂Bt) /A ∂ t is a decreasing function of t>0 and the following relations hold:

≤ Bh ≤ b(m−1) B (12) A (∂Bt) A(∂ t ) e A (∂ t) , ≤ Bh ≤ b(m−1) B V (Bt) V ( t ) e V ( t) .

By the co-area formula, these imply that

⎧ ∗ ∗ p p b(m−1) ⎪ (i) φˆ dvol ≤ φ dvolh ≤ e , ⎨⎪ M λ Mh λ,h (ii) |∇φˆ |∈Lp (M) , (13) λ ⎪ −1 |∇ |p−1 ∈ 1 ⎪ (iii) r(x) φλ φλ L (M), ⎩ p−1 (iv) φλ|∇φλ| dvol = o (R), as R → +∞. BR

Here dvolh stands for the Riemannian measure on Mh. Now, by Laplacian comparison, [10], assumption (4) yields

(m − 1)eb Δr ≤ r pointwise on M \ cut(o) and weakly on all of M. This means that (m − 1)eb (14) − ∇r, ∇η dvol ≤ η dvol, M M r ≤ ∈ 1,2 ≤ ∈ ∞ for all 0 η Wc (M). Let 0 ξ Cc (M) to be chosen later and apply (14) with − ˆ |  |p−2  η = ξφλ ϕλ(r) ϕλ(r),

thus obtaining  |  |p−2 ∇ ∇ ˆ (15) ϕλ(r) ϕλ(r) r, ξφλ dvol M − b ≤− |  |p−2 −  (m 1)e  ˆ ϕλ(r) (p 1)ϕλ(r)+ ϕλ(r) ξφλ dvol. M r On the other hand, according to (9), m − 1 ∗− |ϕ (t)|p−2 (p − 1)ϕ(t)+ ϕ (t) = −K(m, p)−pϕp 1(t) λ λ t λ λ

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for all t>0, and inserting into (15) gives ∗ (m − 1)(eb − 1) (16) − K(m, p)−pξφˆp dvol + φˆ ϕ (r)|ϕ (r)|p−2 ξdvol λ λ λ λ r M M p ≤− ∇ ˆ − ˆ  |  |p−2 ∇ ∇ ξ φλ dvol φλϕλ(r) ϕλ(r) r, ξ dvol M M p ≤− ∇ ˆ − ˆ  |  |p−2|∇ | ξ φλ dvol φλϕλ(r) ϕλ(r) ξ dvol. M M

Now choose ξ = ξR such that ξ ≡ 1onBR, ξ ≡ 0onM \ B2R and |∇ξ| < 2/R. Then, taking the limits as R → +∞ in (16) and recalling (13), we obtain p ∗ ∇ ˆ − −p ˆp φλ dvol K(m, p) φλ dvol M M p−1 − m−p m p p − − ≤ − p b − p p−1 (m 1) (m 1) p−1 β (e 1)λ (λ + r ) dvol, M proving that p ˆ ∇φλ M −p (17) ∗ ≤ K (m, p) + C , ˆp 1 M φλ wherewehaveset 2 p −(m−1) p−1 − p (λ + r p−1 ) dvol − m−p m−p b − M (18) C1(m, p, λ, b):=(m 1) p−1 β (e 1) p . p−1 −m λ M (λ + r ) dvol By (12) and computing explicitly the integrals on Rm,weget ∞ b(m−1) p − − A(∂Bt)e p−1 (m 1) (λ + r ) dvol ≤ p − M 0 (λ + t p−1 )m 1 − m m − − − m−p Γ(m p )Γ( p 1) b(m 1) B p = e A(∂ 1)λ p − , p−1 Γ(m 1) where Γ denotes the Euler Gamma function. On the other hand, Bh B V (Bt) ≤ V ( t ) ≤ b(m−1) V ( t) → (19) p m p m e p m 0, λ + t p−1 λ + t p−1 λ + t p−1

as t →∞. Therefore, we can integrate by parts using the co-area formula, apply (3), integrate by parts again and compute explicitly the integrals on Rm,thus obtaining ∞ p d p − −m − −m (λ + r p 1 ) dvol = V (Bt) − (λ + t p 1 ) dt M 0 dt ∞ d p − −m ≥ γ V (Bt) − (λ + t p 1 ) dt 0 dt ∞ d p − −m = γ A(∂Bt) (λ + t p 1 ) dt 0 dt − m m − m Γ(m p )Γ( p ) B p = γA(∂ 1)λ p . p−1 Γ(m)

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Inserting into (18) and (17), it follows that p ˆ ∇φλ M −p (20) ∗ ≤ K (m, p) + C , ˆp 2 M φλ where − p2 b(m−1) − 2 − p 1 − e C (m, p, b):= (m 1) p m p β m−p (eb − 1) . 2 m−p p−1 γ ˆ On the other hand, because of (13), we can use φλ into (1) and get p ˆ ∇φλ M −p (21) p ≥ C . ∗ M ˆp∗ p M φλ Combining (20) and (21) we obtain ∗ ˆp ≥ −1 φλ C3 , M with C p m/p C (m, p, b, C ):= M + Cp C . 3 M K(m, p) M 2 From this latter, using (13), (19), the co-area formula and integrating by parts, it follows that ∗ ∗ ≤ b(m−1) ˆp − p (22) 0 C3e φλ dvol φλ,hdvolh M Mh ∞ ∗ Bh d − p = vM,h(t)V ( t ) ϕλ (t) dt, 0 dt where, by Bishop-Gromov, the function V (B ) v (t):= C eb(m−1) t − 1 M,h 3 Bh V ( t ) is non-increasing. In view of (11), in order to prove (6), it is enough to show that limt→∞ vM,h(t) ≥ 0. By contradiction, suppose there exist positive constants and T such that vM,h(t) ≤− for all t ≥ T . In this assumption, T0 := sup{t

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is decreasing in λ,providedλ 1. Then we can apply the dominated convergence theorem to deduce T 0 d ∗ (24) lim V (Bh) −ϕp (t) dt → ∞ t λ λ + 0 dt T0 ∗ b(m−1) d p ≤ e lim V (Bt) −ϕ (t) dt =0. → ∞ λ λ + 0 dt On the other hand, using (11) and the co-area formula, we have ∞ ∞ ∗ ∗ Bh d − p ≥ B d − p V ( t ) ϕλ (t) dt V ( t) ϕλ (t) dt 0 dt 0 dt ∞ ∗ ≥ B p A(∂ t)ϕλ (t)dt 0 p∗ = φλ (x) dx =1, Rm for all λ>0. Therefore, by (24), ∞ d ∗ (25) lim V (Bh) −ϕp (t) dt ≥ 1. → ∞ t λ λ + T dt Inserting (24) and (25) into (23) we conclude that, up to choosing λ>0 large enough, ∞ ∗ Bh d − p vM,h(t)V ( t ) ϕλ (t) dt < 0, 0 dt ˆ −1 −b(m−1) which contradicts (22). Setting C(m, p, CM ,b):=C3 e ,wehavethus proven the validity of (6). 

Corollary 2 now follows immediately from (6) recalling the explicit form of Cˆ given in (7) and using the next result due to S.-H. Zhu, [14], later quantified by M. Bazanfar´e, [1]. Theorem 3. Let (M,, ) be a complete m-dimensional Riemannian manifold such that M Sect ≥−G(r(x)) on M, where G satisfies (5) for some b>0. Then there exists an explicit constant 0 < ω0 = ω0(b) → 1/2 as b → 0 such that, if

V (Br(o)) ≥ cV (Br), ∀r>0, m and for some c ≥ ω0,thenM is diffeomorphic to R .

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