Normal form of the metric for a class of Riemannian manifolds with ends

TL;DR

This paper establishes the existence of radial normal coordinates for a broad class of manifolds with ends, including asymptotically conical and hyperbolic types, facilitating PDE analysis involving the Laplace-Beltrami operator.

Contribution

It proves the existence of normal coordinates near infinity for manifolds with ends and analyzes their decay rates and conformal relations to the original metrics.

Findings

Normal coordinates exist near infinity for various manifolds with ends.

The decay rate of the metric in these coordinates is characterized.

The new metrics are conformally equivalent to the original metrics at infinity.

Abstract

In many problems of PDE involving the Laplace-Beltrami operator on manifolds with ends, it is often useful to introduce radial or geodesic normal coordinates near infinity. In this paper, we prove the existence of such coordinates for a general class of manifolds with ends, which contains asymptotically conical and hyperbolic manifolds. We study the decay rate to the metric at infinity associated to radial coordinates and also show that the latter metric is always conformally equivalent to the metric at infinity associated to the original coordinate system. We finally give several examples illustrating the sharpness of our results.