A Kazden-Warner type condition and heat kernel estimates on noncompact manifolds

Abstract

Let M be a noncompact manifold with nonnegative Ricci curvature outside a compact set and with nonnegative scalar curvature. First we generalize and extend the famous Kazdan-Warner and Bourguignon-Ezin condition, from compact manifolds to M. Second we show that the Yamabe problem of prescribing constant positive scalar curvature on those M with positive Yamabe invariant does not have a finite energy solution in most cases. We also prove qualitatively sharp two-sided large time estimates of Schrodinger heat kernels including those of the conformal Laplacian with long range scalar curvature.