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Perpetual cutoff method and discrete Ricci curvature bounds with exceptions

Florentin Münch

Abstract

One of the main obstacles regarding Barky Emery curvature on graphs is that the results require a global uniform lower curvature bounds where no exception sets are allowed. We overcome this obstacle by introducing the perpetual cutoff method. As applications, we prove gradient estimates only requiring curvature bounds on parts of the graph. Moreover, we sharply upper bound the distance to the exception set for graphs having uniformly positive Bakry-Emery curvature everywhere but on the exception set.

Perpetual cutoff method and discrete Ricci curvature bounds with exceptions

Abstract

One of the main obstacles regarding Barky Emery curvature on graphs is that the results require a global uniform lower curvature bounds where no exception sets are allowed. We overcome this obstacle by introducing the perpetual cutoff method. As applications, we prove gradient estimates only requiring curvature bounds on parts of the graph. Moreover, we sharply upper bound the distance to the exception set for graphs having uniformly positive Bakry-Emery curvature everywhere but on the exception set.

Paper Structure

This paper contains 7 sections, 5 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Dini Derivatives, Gamma calculus, and semigroups
  4. Perpetual cutoff method
  5. Gradient estimates
  6. Distance bounds
  7. Discussion and outlook

Key Result

Proposition 3.1

Let $F: I \to \mathbb{R}$ be continuous.

Theorems & Definitions (9)

  • Proposition 3.1: Monotonicity theorem and Theorem 3 from hagood2006recovering
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • proof