Lipschitz Bounds and Uniform Convergence for Sequences of Bounded Rough Riemannian Metrics

TL;DR

This paper investigates bounded rough Riemannian metrics, establishing conditions for Lipschitz and uniform bounds on the induced length space, with examples demonstrating the optimality of these conditions.

Contribution

It identifies the weakest conditions on bounded rough Riemannian metrics that guarantee Lipschitz and uniform bounds, advancing understanding of their geometric properties.

Findings

Established conditions for Lipschitz bounds on length spaces

Established conditions for uniform bounds on length spaces

Provided examples showing these conditions are optimal

Abstract

Here we study what we call bounded rough Riemannian metrics $(M,g)$, which are positive definite, symmetric tensors on each tangent space, $T_pM$, which are bounded and measurable as functions in coordinates. This is enough structure to study the length space given by taking the infimum of the length of all piecewise smooth curves connecting points $p,q \in M$. The goal is to find the weakest conditions one can place on $g$ which can guarantee Lipschitz or uniform bounds from above and below. For each condition, an example is given showing that the condition cannot be weakened any further which also explores the geometric intuition.