On Extensions of Myers' Theorem

Abstract

Let M be a compact Riemannian manifold and h a smooth function on M . Let ρh(x) = inf |v|=1 (Ricx(v, v)− 2Hess(h)x(v, v)). Here Ricx denotes the Ricci curvature at x and Hess(h) is the Hessian of h. Then M has finite fundamental group if ∆h − ρh < 0. Here ∆h =: ∆ + 2L∇h is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of Myers’ theorem to manifolds with mostly positive curvature. There is also a similar result for noncompact manifolds. An early result of Myers 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 Berard and Besson[2]). Wu [12] showed that Myers’ theorem holds if the manifold is allowed to have negative curvature on a set of small diameter, while Elworthy and Rosenberg [8] considered manifolds with some negative curvature on a set of small volume, followed by recent work of Rosenberg and Yang [10]. We use a method of Bakry [1] to obtain a result given in terms of the potential kernel related to ρ(x) = inf |v|=1 Ricx(v, v), which gives a quick probabilistic proof of recent results on extensions of Myers’ theorem. Here Ricx denotes the Ricci curvature at x. Let M be a complete Riemannian manifold, and h a smooth real-valued function on it. Assume Ric−2Hess(h) is bounded from below, where Hess(h) ∗Research supported in part by NATO Collaborative Research Grants Programme 0232/87 and by SERC grant GR/H67263. 1991 Mathematical subject classification 60H30,53C21