Heat flow of extrinsic biharmonic maps from a four dimensional manifold with boundary

TL;DR

This paper proves the existence and regularity of solutions to the heat flow of extrinsic biharmonic maps from a four-dimensional manifold with boundary, including behavior near singular times and existence of smooth maps.

Contribution

It establishes the first global weak solution existence and regularity results for this heat flow with boundary conditions, and analyzes singular time behavior.

Findings

Existence of a unique global weak solution

Regularity except at finitely many times

Existence of smooth extrinsic biharmonic maps under Dirichlet conditions

Abstract

Let $(M,g)$ be a four dimensional compact Riemannian manifold with boundary and $(N,h)$ be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from $M$ to $N$ under the Dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. We also discuss the behavior of solution near the singular times. As an immediate application, we prove the existence of a smooth extrinsic biharmonic map from $M$ to $N$ under any Dirichlet boundary condition.