H\"older Compactification for some manifolds with pinched negative curvature near infinity

Abstract

We consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K of M so that the outward normal exponential map off of the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well defined H\"older structure independent of K. The H\"older constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.