Convexity on manifolds without focal points and applications

Abstract

In this article, we study strictly convex functions on Riemannian manifolds without focal points, a broad class of manifolds encompassing all Hadamard manifolds as well as a large collection of manifolds whose sectional curvatures change sign. Using geometrically defined convex functions on such manifolds, we derive interesting consequences such as the continuity of the isoperimetric profile function without conditions on the sectional curvatures; if the manifolds are also K\"ahler, we obtain Steinness as well as a lower bound on the volume growth of metric balls. Our primary applications concern the spectrum of the Laplacian. We prove that the absolutely continuous part of the spectrum contains a certain infinite interval assuming only the existence of a point with respect to which the radial curvatures are nonpositive. This yields a generalization of the corresponding result for Hadamard manifolds. We use the geometry at infinity to give a new construction of a strictly convex function. We then apply this to show that the spectrum is purely absolutely continuous on a class of manifolds for which the horospheres in every direction at a single point have constant mean curvatures (e.g. asymptotically harmonic manifolds, symmetric spaces of noncompact type). Finally, we show the equality of Cheeger's constant and the volume entropy for a broad class of manifolds.