The Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary

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 ∇(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 ∇(t) does not blow up if the dimension of M is less than 4 or if the initial energy of ∇(t) is sufficiently small.