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

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

2008-03-07

arxiv.org/abs/0803.1015v2

Journal of Functional Analysis 255 (2008), pages 2933-2965

Mathematics

Differential Geometry

37 pages

Scientific paper

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.

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