---
abstract: 'We consider a complete noncompact Riemannian manifold $M$ and give conditions on a compact submanifold $K \subset M$ so that the outward normal exponential map off of the boundary of $K$ is a diffeomorphism onto ${M \backslash K}$. We use this to compactify $M$ and show that pinched negative sectional curvature outside $K$ implies $M$ has a compactification with a well defined Hölder structure independent of $K$. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.'
address: 'University of Washington, Seattle, Washington'
author:
- Eric Bahuaud
- Tracey Marsh
title: |
    Hölder Compactification for some\
    manifolds with pinched negative\
    curvature near infinity
---

The Poincaré model of hyperbolic space has a natural geometric compactification – one can compactify by adding the sphere at infinity. Taking this to be a model case for other simply connected manifolds of negative curvature leads to a classical construction made precise in [@Eberlein-ONeill]: Let $M$ be a Cartan-Hadamard manifold, that is, a complete, simply connected Riemannian manifold with nonpositive sectional curvature. Define an equivalence relation on the set of geodesic rays parametrized by arc length by saying that geodesic rays 