Geometric Inequalities on Riemannian and sub-Riemannian manifolds by heat semigroups techniques

TL;DR

This paper reviews how heat semigroup techniques can be applied to derive geometric inequalities on Riemannian and sub-Riemannian manifolds, highlighting their theoretical significance and applications.

Contribution

It provides a comprehensive overview of heat semigroup methods in geometric analysis, emphasizing new inequalities and their derivations in complex geometric settings.

Findings

Derivation of new geometric inequalities using heat semigroups

Applications to Riemannian and sub-Riemannian geometries

Connections between heat flow and geometric properties

Abstract

In those lecture notes, we review some applications of heat semigroups methods in Riemannian and sub-Riemannian geometry. The notes contain parts of courses taught at Purdue University, Institut Henri Poincar\'e, Levico Summer School and Tata Institute.