Rigidity of the Borell-Brascamp-Lieb Inequality on Weighted Riemannian Manifolds
Rongkai Zhang
TL;DR
This paper investigates the rigidity properties of the Borell-Brascamp-Lieb and Brunn-Minkowski inequalities on weighted Riemannian manifolds, extending previous curvature rigidity results to more general settings.
Contribution
It generalizes the curvature rigidity theorem for these inequalities to weighted Riemannian manifolds and presents new rigidity results and open problems.
Findings
Rigidity theorem for Borell-Brascamp-Lieb inequality on weighted manifolds
Extension of curvature rigidity results to weighted settings
New rigidity results for Brunn-Minkowski inequality
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 Krist\'aly (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.
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