Li-Yau sub-gradient estimates and Perelman-type entropy formulas for the heat equation in quaternionic contact geometry

TL;DR

This paper extends Li-Yau gradient estimates and Perelman-type entropy formulas to the quaternionic contact (qc) heat equation on compact qc manifolds, providing new tools for analysis in quaternionic contact geometry.

Contribution

It introduces the first qc versions of Li-Yau gradient estimates and Perelman entropy formulas, advancing the understanding of heat equations in quaternionic contact geometry.

Findings

Established two sub-gradient estimates for the qc heat equation.

Derived two Perelman-type entropy formulas for qc-Einstein manifolds.

Provided an integral sub-gradient estimate for positive solutions.

Abstract

We establish in the present paper two sub-gradient estimates for the quaternionic contact (qc) heat equation on a compact qc manifold of dimension $4n+3$, provided some positivity conditions are satisfied. These are qc versions of the prominent Li-Yau gradient estimate in Riemannian geometry. Another goal of this paper is to get two Perelman-type entropy formulas for the qc heat equation on a compact qc-Einstein manifold of dimension $4n+3$ with non-negative qc scalar curvature (e.g. compact $3$-Sasakian manifold), as well as an integral sub-gradient estimate for the positive solutions of the qc heat equation.