Convexity on Manifolds without Focal points and applications

In this article, we study strict convexity on simply connected Riemannian manifolds without focal points, a class of manifolds which contains not only Hadamard manifolds but also examples of manifolds whose sectional curvatures change sign. Using the geometrically defined convex functions on such manifolds, namely the distance function and the Busemann function, strictly convex functions with various desired additional properties are constructed. From the former, we derive interesting consequences such as the continuity of the isoperimetric profile function without conditions on the sectional curvature and, if the manifolds are also Kähler, 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 is a substantial generalisation of the corresponding result for Hadamard manifolds. In the latter case of Busemann functions, we use the geometry at infinity to give a new construction of a strictly convex function assuming that the Ricci curvature is negative. We then apply this to show that the spectrum is purely absolutely continuous on a class of manifolds for which all the horospheres have the same constant mean curvature. In particular, for a well-studied subclass for which the condition on Ricci curvature is automatically satisfied, we completely determine the spectrum. This construction also enables us to recover the classical result on the spectrum in the case of symmetric spaces of noncompact type and higher rank even though the mean curvatures of horospheres there are not constant.