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Rigidity of the Borell-Brascamp-Lieb Inequality on Weighted Riemannian Manifolds

Rongkai Zhang

Abstract

In this paper we discuss some results regarding the rigidity of the Borell-Brascamp-Lieb inequality and the Brunn-Minkowski inequality. We show a theorem of rigidity on curvature and measure of the Borell-Brascamp-Lieb inequality, a generalisation of the curvature rigidity theorem by Balogh and Kristály (Advances in Mathematics 339: 453-494, 2018, arXiv:1704.04180) to the weighted setting. We present some rigidity results of the Brunn-Minkowski inequality and a few further open problems.

Rigidity of the Borell-Brascamp-Lieb Inequality on Weighted Riemannian Manifolds

Abstract

In this paper we discuss some results regarding the rigidity of the Borell-Brascamp-Lieb inequality and the Brunn-Minkowski inequality. We show a theorem of rigidity on curvature and measure of the Borell-Brascamp-Lieb inequality, a generalisation of the curvature rigidity theorem by Balogh and Kristály (Advances in Mathematics 339: 453-494, 2018, arXiv:1704.04180) to the weighted setting. We present some rigidity results of the Brunn-Minkowski inequality and a few further open problems.

Paper Structure

This paper contains 7 sections, 8 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rigidity of the Borell-Brascamp-Lieb inequality
  4. The Brunn-Minkowski Inequality and its Rigidity
  5. Rigidity in the null curvature bound
  6. Rigidity in the positive and negative curvature bounds
  7. Further Problems

Key Result

Lemma 1

On a weighted Riemannian manifold $(M,g,m = e^{-\psi(x)}\mathrm{vol}_g)$ satisfying $\mathrm{Ric}_{m,N} \geq Kg$, the weighted Jacobian $J^\psi_t(x) := e^{\psi(x) - \psi(\mathcal{F}_t(x))} \|(D\mathcal{F}_t(x))\|$ defined along the geodesic $\gamma(t):= \mathcal{F}_t(x)$, where $\mathcal{F}_t(x)=\ex $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • ...and 7 more