On extensions of Myers' theorem

Let M be a compact Riemannian manifold and h a smooth function on M. Let h (x) = inf jvj=1 (Ric x (v; v) ? 2Hess(h) x (v; v)). Here Ric x denotes the Ricci curvature at x and Hess(h) is the Hessian of h. Then M has nite fundamental group if h ? h < 0. Here h =: + 2L rh is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of Myers' theorem to mani-folds 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 nite fundamental group. See e.g. 9]. Since then eeorts have been made to get the same type of result but to allow a little bit of negativity of the curvature (see B erard and Bessonn2]). 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 (x) = inf jvj=1 Ric x (v; v), which gives a quick probabilistic proof of recent results on extensions of Myers' theorem. Here Ric x 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)