Uniformly Regular and Singular Riemannian Manifolds

TL;DR

This paper investigates the properties of uniformly regular and singular Riemannian manifolds, highlighting their importance for Sobolev space solutions to parabolic equations on non-compact manifolds, and provides examples of such manifolds.

Contribution

It offers a comprehensive analysis of uniformly regular and singular manifolds and identifies classes suitable for parabolic PDE solution theories.

Findings

Characterization of uniformly regular and singular manifolds

Identification of large families of admissible manifolds

Connection between manifold classes and Sobolev space solutions

Abstract

A detailed study of uniformly regular Riemannian manifolds and manifolds with singular ends is carried out in this paper. Such classes of manifolds are of fundamental importance for a Sobolev space solution theory for parabolic evolution equations on non-compact Riemannian manifolds with and without boundary. Besides pointing out this connection in some detail we present large families of uniformly regular and singular manifolds which are admissible for this analysis.