Compactness of Sequences of Warped Product Length Spaces

TL;DR

This paper investigates conditions under which sequences of warped product length spaces exhibit compactness and convergence, providing criteria on warping functions that ensure Lipschitz bounds and well-defined limits.

Contribution

It offers new theorems linking warping function conditions to compactness and convergence of warped product length spaces, with illustrative examples.

Findings

Conditions on warping functions imply compactness of the sequence

Criteria for Lipschitz bounds on the sequence of distances

Examples demonstrating necessity of hypotheses

Abstract

If we consider a sequence of warped product length spaces, what conditions on the sequence of warping functions implies compactness of the sequence of distance functions? In particular, we want to know when a subsequence converges to a well defined metric space on the same manifold with the same topology. What conditions on the sequence of warping functions implies Lipschitz bounds for the sequence of distance functions and/or the limiting distance function? In this paper we give answers to both of these questions as well as many examples which elucidate the theorems and show that our hypotheses are necessary.