Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Brian Allen, Bernardo Falcao, Harry Pacheco, Bryan Sanchez

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.

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

Abstract

Here we study what we call bounded rough Riemannian metrics , which are positive definite, symmetric tensors on each tangent space, , 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 . The goal is to find the weakest conditions one can place on 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.
Paper Structure (16 sections, 18 theorems, 144 equations)

This paper contains 16 sections, 18 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Conformal Length Spaces
  4. Bounded Rough Riemannian Metrics
  5. Uniform Convergence
  6. Minimizing Classes of Curves
  7. Conformal Metric Space Examples
  8. Examples of Sequences of Metric Spaces
  9. Blow Up Examples
  10. Blow up on a Disappearing Center Square
  11. Blow up on a Center Line
  12. Shortcut Examples
  13. Shortcut on a Disappearing Center Square
  14. Shortcut Along A Curve of Hausdorff $1$ Measure Zero
  15. Shortcut Along A Rectangle Containing The Center Line
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $g_1$ be a bounded rough Riemannian metric, $g_0$ a Riemannian metric, and $M$ a smooth, connected, compact manifold. If there exists a $U \subset M$ so that $\mathcal{H}_{g_0}^1(M \setminus U) = 0$ and there exists a $c > 0$ so that then

Theorems & Definitions (41)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • ...and 31 more