Gradient Estimates and Sobolev Inequality

Transcription

1 Gradient Estimates and Sobolev Inequality Jiaping Wang University of Minnesota ( Joint work with Linfeng Zhou) Conference on Geometric Analysis in honor of Peter Li University of California, Irvine January 15, 2012 (University of Minnesota) Gradient Estimates and Sobolev Inequality 1 / 30

2 1 Introduction 2 Function case 3 Differential form case (University of Minnesota) Gradient Estimates and Sobolev Inequality 2 / 30

3 Setup Notations (M n, g), a compact oriented Riemannian manifold without boundary. Hodge Laplacian : A p (M) A p (M), acting on the space of smooth p-forms A p (M) on M, is defined by = dδ δd. As usual, d is the exterior differential operator and δ the adjoint of d with respect to the L 2 inner product on A p (M). Eigenvalues of are denoted by {0 λ 1... λ k... } with a corresponding orthonormal basis of eigenforms {φ i } i=1. (University of Minnesota) Gradient Estimates and Sobolev Inequality 3 / 30

4 Projection Obviously, {φ i } bp i=1 is a basis of the space Hp (M) of p-harmonic forms, where b p is the p-th Betti number of M. Projection of a p-form ω onto H p (M) is then given by with For function f, b p P(ω) = a i φ i a i = M i=1 ω, φ i. P(f ) = 1 f. V (M) M (University of Minnesota) Gradient Estimates and Sobolev Inequality 4 / 30

5 Sobolev Inequality There exists a constant c > 0 such that c ( M ) n 2 ω P(ω) 2n n n 2 M { dω 2 + δω 2 } for all smooth p form ω on M, where P(ω) denotes the projection of ω on to the space of harmonic p forms. Objective: obtain an explicit estimate of the constant c in terms of the geometry of M. (University of Minnesota) Gradient Estimates and Sobolev Inequality 5 / 30

6 Gradient Estimate, function case For acting on functions, denoted its eigenvalues by {0 = λ 0 < λ 1... λ k... }, and a corresponding orthonormal basis of eigenfunctions {φ i } i=0. Let (M n, g) be a closed Riemannian manifold with Ricci curvature lower bound (n 1)K, where K 0 is a constant. Let and E k = {v = k k a i φ i, ai 2 = 1} i=1 i=1 β k = max v E k max v (x). x M Lemma v 2 + (λ k + (n 1)K) v 2 (λ k + (n 1)K) β 2 k. (University of Minnesota) Gradient Estimates and Sobolev Inequality 6 / 30

7 Consequences Theorem There exists an explicit constant c(k, d, V, n), where d and V are the diameter and volume of M respectively, such that (1) v 2 cλ n+2 2 k, v 2 cλ n 2 k, for all v E k. (2) For all k 1, (3) The function H(x, y, t) given by λ k c 1 k 2 n. H(x, y, t) = 1 V + e λkt φ k (x) φ k (y) k=1 is the heat kernel of M. Moreover, for all t > 0, H(x, y, t) 1 V c t n 2. (University of Minnesota) Gradient Estimates and Sobolev Inequality 7 / 30

8 Proof Applying the gradient estimate, we have β k c(k, d, V, n) λ n 4 k For each x M, there exists an orthogonal matrix (a ij ) k k such that ψ j (x) = 0 for j = n + 1,, k, where ψ j = k i=1 a ij φ i. Now k φ i 2 (x) = i=1 n j=1 ψ j 2 (x) n β k c λ n+2 2 k. (University of Minnesota) Gradient Estimates and Sobolev Inequality 8 / 30

9 Proof, continued Integrating the inequality with respect to x, one concludes which implies λ 1 + λ λ k c 2 λ n+2 2 k, λ k c 3 k 2/n It is straightforward to see that H(x, y, t) 1 V e λkt φ k (x) φ k (y) k=1 c t n 2. (University of Minnesota) Gradient Estimates and Sobolev Inequality 9 / 30

10 Varopoulos Result The Sobolev inequality follows from a result of Varopoulos. Theorem If H(x, y, t) 1 V A t n 2, then the following Sobolev inequality holds with some c = c(a). c ( M ) n 2 f 2n n n 2 for all smooth function f on M with M f = 0. M f 2 (University of Minnesota) Gradient Estimates and Sobolev Inequality 10 / 30

11 Another Consequence c 1 k βk c 2 k. That is, for each k, there exist a 1, a 2,, a k with k i=1 a2 i = 1 such that k max a i φ i c k. x M i=1 (University of Minnesota) Gradient Estimates and Sobolev Inequality 11 / 30

12 Remarks The arguments work for M with non-empty boundary by some modifications. For the Dirichlet boundary conditions, the constants involve the mean curvature of the boundary, while for the Neumann boundary conditions, the constants involve both the second fundamental form and the size of the rolling ball along the boundary. The arguments rely crucially on a result of Li and Yau, which says that λ 1 c(n, d, K). One can in fact strengthen the gradient estimate so that it also incorporates Li and Yau s result. Indeed, one has for v E k and β > β k, Lemma v 2 (β v) 2 (n 1)( 2(n 1)K + (2k + 1)β λ k β β k ). (University of Minnesota) Gradient Estimates and Sobolev Inequality 12 / 30

13 Remarks, continued The Sobolev inequality can also be obtained from Li and Yau s famous heat kernel estimates. However, it relies on both Li and Yau s estimate of λ 1 and the existence of heat kernel. Historically, Yau and Croke have estimated the isoperimetric constant in terms of the geometry of M, which leads to the Sobolev inequality. The lower bound estimate of λ k was first established by Cheng and Li using the Sobolev inequality. Later, it was also derived by Li and Yau from their heat kernel estimates. (University of Minnesota) Gradient Estimates and Sobolev Inequality 13 / 30

14 Remarks, continued Polya conjecture: For all k 1, λ k c n ( k V ) n/2, where 0 < λ 1 < λ 2 are the Dirichlet eigenvalues of a compact Euclidean domain Ω R n, and c n = (2π) 2 /(ω n ) 2/n, ω n the volume of the unit ball in R n. Theorem (Li-Yau) k λ i i=1 n n + 2 c n k ( ) k n/2. V (University of Minnesota) Gradient Estimates and Sobolev Inequality 14 / 30

15 Form case, Difficulty Using the well-known Bochner-Weitzenbock formula, one can directly apply the preceding proof of the function case to the Hodge Laplacian acting on the smooth p forms on M. However, the resulting estimates will depend on the bounds of the covariant derivative of the curvature tensor of M. On the other hand, by working with the eigenforms, but not their covariant derivatives, Li had obtained the following interesting results 30 years ago. (University of Minnesota) Gradient Estimates and Sobolev Inequality 15 / 30

16 Work of Li Let E k = {v = k k a i φ i, ai 2 = 1}. i=1 i=1 Theorem For a compact manifold with its curvature operator Rm K, the following estimates hold for k c(k, b p, n). (1) For all v E k, v (x) c λ n 1 2 k. (2) Consequently, λ k c k 1 n 1. Note that in both estimates, the order is not sharp. (University of Minnesota) Gradient Estimates and Sobolev Inequality 16 / 30

17 Gradient estimate for forms Theorem Let (M n, g) be a closed manifold with curvature bound Rm K. Then for any form v E k, it satisfies the estimate v 2 + (λ k + 1) v 2 c (λ k + 1) n 2 +1, where c = c(n, V, d, K) is an explicit constant depending on the dimension n, volume V, diameter d and the curvature bound K. (University of Minnesota) Gradient Estimates and Sobolev Inequality 17 / 30

18 Proof Consider the function f = ω 2 + A ω 2, where A 1 is a fixed constant and ω a smooth p-form. Lemma For m 1, M f m 1 f 2 where c = 2nK(K + 2) + 18 K 2. M ( ω, ω + A ω, ω ) f m 1 c m 2 M f m, (University of Minnesota) Gradient Estimates and Sobolev Inequality 18 / 30

19 Proof, continued Proof of the gradient estimate: For each m 1, let I m = max f 2m, where f = v 2 + A v 2, A = λ k + K + 1, and the maximum is taken over all v E k. Note that where k i=1 b2 i 1. The lemma implies v = M M k λ i a i φ i = λ k i=1 k b i φ i, i=1 f 2m 1 f (2 λ k + c m 2 ) I m, (1) where c 1 = 8nK(K + 2) + 72 K 2. (University of Minnesota) Gradient Estimates and Sobolev Inequality 19 / 30

20 Proof, continued On the other hand M f 2m 1 f = 2m 1 m 2 Applying the Sobolev inequality, we get ( M f 2mβ ) 1 β M c m (λk + c m 2 ) I m f m 2. (2) Since this is true for all v E k, we maximize the left hand side over v and conclude (I βm ) 1 βm ( c m (λ k + c m 2 ) ) 1 m (I m ) 1 m for all m 1, where β = n n 1. The result then follows from iteration. (University of Minnesota) Gradient Estimates and Sobolev Inequality 20 / 30

21 Consequences Using a result of Mantuano that λ bp+1 c(n, V, d, K), we have There exists a constant c(k, d, V, n) such that (1) φ k c (λ k + 1) n+2 4 and φ k c (λ k + 1) n 4. (2) For all k > b p, λ k c 1 k 2 n. (3) The tensor H p (x, y, t) is the heat kernel of, where Moreover, for all t > 0, H p (x, y, t) = e λkt φ k (x) φ k (y) k=1 b(p) H p (x, y, t) φ k (x) φ k (y) c t n 2 k=1 (University of Minnesota) Gradient Estimates and Sobolev Inequality 21 / 30

22 Consequences, continued Let (M n, g) be a closed manifold with curvature bound Rm K. Then there exists a constant c(k, d, V, n) such that for v E k. d v c λ n+4 4 k In particular, this gives a Hessian estimate for eigenfunctions on M. (University of Minnesota) Gradient Estimates and Sobolev Inequality 22 / 30

23 Sobolev Inequality Theorem The following Sobolev inequality holds. c ( M ) n 2 ω P(ω) 2n n n 2 M { dω 2 + δω 2 } for all smooth p form ω on M, where P(ω) denotes the projection of ω on to the space H p (M) of harmonic p forms. (University of Minnesota) Gradient Estimates and Sobolev Inequality 23 / 30

24 Rumin s result Let A : L 2 (X, µ) L 2 (X, µ) be a non-negative self-adjoint operator. Let V = L 2 (X, µ) (kera). Suppose M(t) = t L(s) ds <, where L(t) = e ta Π V 1, and Π V : L 2 (X, µ) V the orthogonal projection. Then for f V, X N ( f 2 (x)) ln 2, 4D(f ) where D(f ) = Af, f L 2 and N(y) = y M 1 (y). (University of Minnesota) Gradient Estimates and Sobolev Inequality 24 / 30

25 Special case of Rumin s result If then and The inequality becomes c ( X L(t) c t n/2, M(t) = c t 2 n 2 M 1 (t) = c t 2 2 n. f 2n ) n 2 n 2 (x) dµ n X f Af dµ. (University of Minnesota) Gradient Estimates and Sobolev Inequality 25 / 30

26 Proof of Rumin s result For f V, Claim: On {x X : f 2 (x) 4 M 2 (t) D(f )}, Indeed, But, So f (x) 4 (I e ta/2 )f 2 (x). e ta/2 f D(f ) 1/2 A 1 e ta Π V 1,. A 1 e ta Π V = t e sa Π V ds. e ta/2 f D(f ) 1/2 e sa Π V 1, ds D(f ) 1/2 M(t). t (University of Minnesota) Gradient Estimates and Sobolev Inequality 26 / 30

27 Proof, continued By the claim, f 2 (x) X 4D(f ) /M 1( f 2 (x)) 4D(f ) 1 (I e ta/2 )f 2 dt 2 D(f ) 0 t 2 1 = (1 e tλ/2 ) 2 d Π λ f, f dt D(f ) 0 t 2 = ln 2. (University of Minnesota) Gradient Estimates and Sobolev Inequality 27 / 30

28 Remarks Compare with Li s result. While our estimates are sharp in terms of the order of k, the constant however depends on both upper and lower bounds of the curvature operator. A natural issue is to see if this dependency can be improved. The results can be extended to the case that M has nonempty boundary. It is desirable to get a lower bound estimate of λ bp+1 of Mantuano by a more analytical proof. (University of Minnesota) Gradient Estimates and Sobolev Inequality 28 / 30

29 Remarks, continued It is unclear to me how to establish Sobolev type inequalities on complete, noncompact manifolds for differential forms. For the function case, this issue has been largely settled through the heat kernel estimates of Li and Yau. (University of Minnesota) Gradient Estimates and Sobolev Inequality 29 / 30

30 Thank you for your attention. (University of Minnesota) Gradient Estimates and Sobolev Inequality 30 / 30