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 $\nabla(t)$ 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.
The Yang-Mills heat equation with finite action in three dimensions
The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over 

 
 
 
 R
 
 3
 
 \mathbb {R}^3
 

 and over a bounded open convex set in 

 
 
 
 R
 
 3
 
 \mathbb {R}^3
 

. The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.
On the Geodesical Connectedness for a Class of Semi-Riemannian Manifolds
We prove a variational principle for geodesics on a semi-Riemannian manifold M of arbitrary index k and possessing k linearly independent Killing vector fields that generate a timelike distribution on M. Using such a principle and a suitable completeness condition for M, we prove some existence and multiplicity results for geodesics joining two fixed points of M.