On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smooth function on M. Let ph(x) = inf|υ|−1(Ricx(υ,υ)−2Hess(hx(υ,υ)). Here Ricx denotes the Ricci curvature at x and Hess(h) is the Hessian of h. Then M has finite fundamental group if Δh−ph <0. Here Δh =:Δ+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.