The present dissertation discusses three connected subjects. Firstly, we establish a version of the Hopf boundary point lemma for sections of a vector bundle over a manifold M with nonempty boundary. This result may be viewed as a counterpart to Richard Hamilton's tensor maximum principle. Secondly, we prove the Li-Yau-Hamilton estimate for the heat equation on M. Such estimates are typically used to derive monotonicity formulas related to geometric flows. Thirdly, we establish bounds for a solution (t) of the Yang-Mills heat equation in a vector bundle over M. 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.