On extensions of Myers theorem

Transcription

1 On extensions of yers theorem Xue-ei Li Abstract Let be a compact Riemannian manifold and h a smooth function on. Let ρ h x = inf v =1 Ric x v, v 2Hessh x v, v. Here Ric x denotes the Ricci curvature at x and Hessh is the Hessian of h. Then has finite fundamental group if h ρ h <. Here h =: + 2L h is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of yers theorem to manifolds with mostly positive curvature. There is also a similar result for noncompact manifolds. An early result of yers says a complete Riemannian manifold with Ricci curvature bounded below by a positive number is compact and has finite fundamental group. See e.g. [9]. Since then efforts have been made to get the same type of result but to allow a little bit of negativity of the curvature see Bérard and Besson[2]. Wu [12] showed that yers theorem holds if the manifold is allowed to have negative curvature on a set of small diameter, while lworthy and Rosenberg [8] considered manifolds with some negative curvature on a set of small volume, followed by recent work of Rosenberg and Yang [1]. We use a method of Bakry [1] to obtain a result given in terms of the potential kernel related to ρx = inf v =1 Ric x v, v, which gives a quick probabilistic proof of recent results on extensions of yers theorem. Here Ric x denotes the Ricci curvature at x. Let be a complete Riemannian manifold, and h a smooth real-valued function on it. Assume Ric 2Hessh is bounded from below, where Hessh Research supported in part by NATO Collaborative Research Grants Programme 232/87 and by SRC grant GR/H athematical subject classification 6H3,53C21 1

2 is the hessian of h. Denote by h the Bismut-Witten Laplacian with probabilistic sign convention defined by: h = + 2L h on CK the space of smooth differential forms with compact support. Here L h is the Lie derivative in direction of h. Then the closure of h is a negative-definite self-adjoint differential operator on L 2 functions or L 2 differential forms with respect to e 2h dx for dx the standard Lebesgue measure on. We shall use the same notation for h and its closure. By the spectral theorem there is a heat semigroup Pt h satisfying the following heat equation: ux, t t = 1 2 h ux, t. We shall denote by Pt h φ the solution with initial value φ. For clarity, we also use P h,1 t for the corresponding heat semigroup for one forms. Then for a function f in CK, dp h t f = P h,1 t df. 1 On there is a h-brownian motion {F t x : t }, i.e. a path continuous strong arkov process with generator 1 2 h for each starting point x. For a fixed point x, we shall write x t = F t x. Then Pt h fx = ff t x for all bounded L 2 functions. Let {W h t, t } be the solution flow to the following covariant equation along h-brownian paths {x t }: D t W h = 1 2 Ric x t W h t v, # + Hesshxt W h t v, #, t v W h v = v, v T x. 2 Here # stands for the adjoint. The solution flow Wt h flow. Let φ be a bounded 1-form, then for x, and v T x is called the Hessian φw h t v = P h,1 t φv. 3 if Ricci 2 Hessh is bounded from below. See e.g. [4] and [5]. Formula 3 gives the following estimates on the heat semigroup: P h,1 t φ φ Wt h. 4 2

3 Let ρ h x = inf v =1 Ric x v, v 2Hesshxv, v and write ρ for ρ h if h =. Then covariant equation 2 gives: Wt h e 1 t 2 ρh x sds 5 as in [4]. Let P ρh t 1 2 ρh. Then be the L 2 semigroup generated by the Schrödinger operator 1 2 h [ P ρh t fx = fx t e 1 t ] 2 ρh x sds by the Feyman-Kac formula. So equation 5 is equivalent to W h t P ρh t 1. Let U f be the corresponding potential kernel defined by: [ Ufx = fx t e 1 t ] 2 ρh x sds dt. Following Bakry s paper [1], we have the following theorem: Theorem 1 Let be a complete Riemannian manifold with Ric 2Hessh bounded from below. Suppose sup U1x < 6 x K for each compact set K. Then has finite h-volumei.e. e2hx dx <, and finite fundamental group. Proof: We follow [1]. Let f C K, then Hf = lim t P t f is an L 2 harmonic function. Assume h-vol =, then Hf =. We shall prove this is impossible. Let f, g C K, then: = = Pt h f fge 2h dx t s P s h f t ge 2h dx < P h s f, g > e 2h dxds 3

4 f f t g sup x supg c f g L 1. Here c = sup x supg [U1x] = sup x supg supg denotes the support of g. Ws h ds W h s ds e 2h dx, g L 1 e t ρh F sxds dt, and Next take f = h n, for h n an increasing sequence of smooth functions approximating 1 with h n 1 and h n 1, see e.g. [1]. n Then P h t h n h n ge 2h dx c 1 n g L 1. First let t go to infinity, then let n to obtain: ge 2h dx. This gives a contradiction with a suitable choice of g. So we conclude h- vol <. Let p: be the universal covering space for with induced Riemannian metric on. For p x = x, let { F t x, t } be the horizontal lift of {F t x} to. Denote by Ric the Ricci curvature on, h the lift of h to, and ρ h the corresponding lower bound for Ric 2Hess h. Then the induced { F t x, t } is a h-brownian motion on. See e.g. [4]. Note also ρ h satisfies sup x K e 1 2 t ρ h F s xds dt = sup e 1 t 2 ρh F sxds dt x p K for any K compact. The same calculation as above will show that has finite h-volume, therefore p is a finite covering and so has finite fundamental group. Let rx be the Riemannian distance between x and a fixed point of and take h to be identically zero: Corollary 2 Let be a complete Riemannian manifold with Ric x > n 1 n 1 r 2 x, when r > r 7 4

5 for some r >. Then the manifold is compact if sup x K U1x < for each compact set K. Proof: This is a consequence of the result of [3]: A complete Riemannian manifold with 7 has infinite volume. See also [11]. For another extension of yers compactness theorem, see [3] where a diameter estimate is also obtained. In the following we shall assume is compact and get the following corollary: Corollary 3 Let be a compact Riemannian manifold and h a smooth function on it. Then has finite fundamental group if h ρ h <. Proof: Let λ be the minimal eigenvalue of h ρ h. Then 1 lim t t sup log e 1 t 2 ρh x sds λ <. See e.g. [7]. Thus there is a number T such that if t T, sup e 1 t 2 ρh x sds e λt. Therefore sup T <. e 1 t 2 ρh x sds dt e 1 2 inf x [ρ h x]t dt + e 1 t 2 ρh x sds dt T The result follows from theorem 1. Let d h = e h de h. It has adjoint δ h = e h δe h on L 2, dx. Let h be the Witten Laplacian defined by: h = d h + δ h 2. By conjugacy of h on L 2, dx with h on L 2, e 2h dx, the condition h ρ h < becomes: h ρ h < 5

6 on L 2, dx. On the other hand, h = dh 2 h. See e.g. [6]. This gives: a compact manifold has finite fundamental group if on L 2, dx. dh 2 h ρ h < Corollary 3 leads to the following theorem from [1]: Let N = N K, D, V, n be the collection of n-dimensional Riemannian manifolds with Ricci curvature bounded below by K, diameter bounded above by D, and volume bounded below by V. Corollary 4 Rosenberg& Yang Choose R >. There exists a = an, R such that a manifold N with vol{x : ρx < R } < a has finite fundamental group. Here vol denotes the volume of the relevant set. Proof: Let h = in corollary 3. Then under the assumptions in the corollary, ρ < according to [8]. The conclusion follows from corollary 3. Acknowledgement: The author is grateful to Professor D. lworthy and Professor S. Rosenberg for helpful comments and encouragement. References [1] D. Bakry. Un critére de non-explosion pour certaines diffusions sur une varíeté Riemannienne compléte. C. R. Acad. Sc. Paris, t. 33, Série I1:23 26, [2] P. Bérard and G. Besson. On the number of bound states and estimates on some geometric invariants. Springer Lect. Notes, [3]. Cheeger,. Gromov, and. Taylor. Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. of Diff. Geom., 17:15 53,

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