The Li-Yau-Hamilton Estimate and the Yang-Mills Heat Equation on Manifolds with Boundary

TL;DR

This paper extends the Li-Yau-Hamilton estimate to manifolds with boundary and applies it to derive bounds for solutions of the Yang-Mills heat equation, ensuring curvature control under certain conditions.

Contribution

It establishes the Li-Yau-Hamilton estimate on manifolds with boundary and applies it to bound Yang-Mills heat flow solutions, linking geometric analysis with gauge theory.

Findings

Li-Yau-Hamilton estimate proven for manifolds with boundary

Curvature of Yang-Mills solutions remains bounded under specific conditions

Results prevent curvature blow-up in low dimensions or with small initial energy

Abstract

The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold $M$ with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution $\nabla(t)$ of the Yang-Mills heat equation in a vector bundle over $M$. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply that the curvature of $\nabla(t)$ does not blow up if the dimension of $M$ is less than 4 or if the initial energy of $\nabla(t)$ is sufficiently small.