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Optimal Transport and Generalized Ricci Flow

Eva Kopfer, Jeffrey Streets

Abstract

We prove results relating the theory of optimal transport and generalized Ricci flow. We define an adapted cost functional for measures using a solution of the associated dilaton flow. This determines a formal notion of geodesics in the space of measures, and we show geodesic convexity of an associated entropy functional. Finally, we show monotonicity of the cost along the backwards heat flow, and use this to give a new proof of the monotonicity of the energy functional along generalized Ricci flow.

Optimal Transport and Generalized Ricci Flow

Abstract

We prove results relating the theory of optimal transport and generalized Ricci flow. We define an adapted cost functional for measures using a solution of the associated dilaton flow. This determines a formal notion of geodesics in the space of measures, and we show geodesic convexity of an associated entropy functional. Finally, we show monotonicity of the cost along the backwards heat flow, and use this to give a new proof of the monotonicity of the energy functional along generalized Ricci flow.
Paper Structure (6 sections, 15 theorems, 33 equations)

This paper contains 6 sections, 15 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Wasserstein distance monotonicity for generalized Ricci flow
  4. Adapted cost for generalized Ricci flow
  5. Geodesic entropy convexity
  6. Cost monotonicity

Key Result

Lemma 2.3

Given $(g, H)$ as above, one has where for a $(k-1)$-form $\alpha$ and $k$-form $\beta$, the notation $\left<\alpha, \beta\right>$ denotes the $1$-form uniquely defined by $\left<\alpha,\beta\right>(X) = \left<\alpha, i_X \beta\right>$.

Theorems & Definitions (17)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4: Streetsscalar
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Corollary 3.6
  • ...and 7 more