THE FORMATION OF SINGULARITIES IN THE HARMONIC MAP HEAT FLOW WITH BOUNDARY CONDITIONS
THE FORMATION OF SINGULARITIES IN THE HARMONIC MAP HEAT FLOW WITH BOUNDARY CONDITIONS
Let $M$ be a compact manifold with boundary and $N$ be compact manifold without boundary. Let $u(x,t)$ be a smooth solution of the harmonic heat equation from $M$ to $N$ with Dirichlet or Neumann condition. Suppose that $M$ is strictly convex, we will prove a monotonicity formula for $u$. Moreover, …