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

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.

37 pages

References in corpus (1)

• On the Holonomy of the Coulomb Connection over Manifolds with Boundary