Sharp differential estimates of Li-Yau-Hamilton type for positive $(p,p)$-forms on K\"ahler manifolds

TL;DR

This paper establishes sharp differential Harnack estimates for positive solutions of the Hodge-Laplacian heat equation on K"ahler manifolds, extending to coupled flows and matrix estimates.

Contribution

It introduces new sharp differential estimates for positive $(p,p)$-forms under heat flow on K"ahler manifolds, including coupled K"ahler-Ricci flow scenarios.

Findings

Proved preservation of positivity for $(p,p)$-forms under heat flow.

Established sharp Li-Yau-Hamilton type differential Harnack inequalities.

Derived matrix differential Harnack estimates for K"ahler-Ricci and Ricci flows.

Abstract

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a K\"ahler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with the K\"ahler-Ricci flow and some interpolating matrix differential Harnack type estimates for both the K\"ahler-Ricci flow and the Ricci flow.