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/H67263. 1991 athematical subject classification 6H3,53C21 1
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
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
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
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
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, 1986. [2] P. Bérard and G. Besson. On the number of bound states and estimates on some geometric invariants. Springer Lect. Notes, 1986. [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, 1982. 6
[4] K. D. lworthy. Geometric aspects of diffusions on manifolds. In P. L. Hennequin, editor, cole d té de Probabilités de Saint-Flour XV-XVII, 1985, 1987. Lecture Notes in athematics, volume 1362, pages 276 425. Springer-Verlag, 1988. [5] K. D. lworthy. Stochastic flows on Riemannian manifolds. In. A. Pinsky and V. Wihstutz, editors, Diffusion processes and related problems in analysis, volume II. Birkhauser Progress in Probability, pages 37 72. Birkhauser, Boston, 1992. [6] K.D. lworthy and S. Rosenberg. The Witten Laplacian on negatively curved simply connected manifolds. To appear in Tokyo J. ath. [7] K.D. lworthy and S. Rosenberg. Generalized Bochner theorems and the spectrum of complete manifolds. Acta Appl. ath., 12:1 33, 1988. [8] K.D. lworthy and S. Rosenberg. anifolds with wells of negative curvature. Invent. ath., 13:471 495, 1991. [9] S. Gallot, D Hulin, and J. Lafontaine. Riemannian geometry, second edition. Springer-Verlag, 199. [1] S. Rosenberg and D. Yang. Bounds on the fundamental group of a manifold with almost non-negative Ricci curvature. To appear in J. ath. Soc. Japan. [11] H. WU. Subharmonic functions and the volume of a noncompact manifold. In Differential geometry, a symposium in honour of anfred do Carmo. Longman scientific & technical, 1991. [12] J-Y Wu. Complete manifolds with a little negative curvature. Amer. J. ath., 113:387 395, 1991. Address: athematics Institute, University of Warwick, Coventry CV4,7AL U.K. e-mail: xl@maths.warwick.ac.uk.bitnet 7