Heat Kernels on Riemannian Polyhedra and Heat Flows into NPC Manifolds

Abstract

We extend the results of Eell-Sampson to show that for a continuous initial map, f, with bounded pointwise energy from a flat, compact, admissible polyhedron to a smooth compact Riemannian manifold with non-positive sectional curvature, there exists a heat flow beginning at f that converges uniformly and in energy to a harmonic map. We show that this heat flow is in $C^{1+\alpha,1+\beta}$, $\alpha, \beta>0$, on open sets bounded away from the (n-2)-skeleton, satisfies a natural balancing condition on the (n-1)-skeleton, and solves the harmonic map heat flow equation pointwise on the interior of top-dimensional simplexes. We develop Gaussian-type estimates for the gradient of heat kernel on a flat, compact, admissible polyhedron, and methods to address existence and regularity of partial differential equations on admissible polyhedra