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A Sudakov Decomposition in Riemannian Manifolds with Positive Curvature

Zhengyao Huang

Abstract

In this paper, we study Monge's problem on Riemannian manifolds $(M, g)$ with positive sectional curvature. Assuming that the source and target measures are absolutely continuous with respect to the Riemannian volume measure, we generalize a variational method from the Euclidean setting to establish the existence of a transport density and an explicit disintegration of measures along optimal rays. These results extend the approach of Bianchini-Caravenna to the Riemannian context.

A Sudakov Decomposition in Riemannian Manifolds with Positive Curvature

Abstract

In this paper, we study Monge's problem on Riemannian manifolds with positive sectional curvature. Assuming that the source and target measures are absolutely continuous with respect to the Riemannian volume measure, we generalize a variational method from the Euclidean setting to establish the existence of a transport density and an explicit disintegration of measures along optimal rays. These results extend the approach of Bianchini-Caravenna to the Riemannian context.
Paper Structure (8 sections, 17 theorems, 133 equations, 1 figure)

This paper contains 8 sections, 17 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Monge problem
  3. Main Theorem
  4. Disintegration of transport sets
  5. Elementary structure of the transport set
  6. Sheaf sets and cylinder sets
  7. Approximation using Jacobi fields
  8. Regularity of the divergence

Key Result

Theorem 1

(Explicit disintegration of the transport set) For every $\varphi\in L^1(M,\mathrm{Vol}_g)$, where $a(y)$ and $b(y)$ are the endpoints of the maximal transport ray through $y$.

Figures (1)

  • Figure 1: schematic drawing of a sheaf set: the transport rays are mutually disjoint geodesics (red). They pierce the hypersurface slice (gray). The set $Z$ is a subset of a level set of $u$, and $Z$ is a compact base set.

Theorems & Definitions (38)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 28 more