Existence of expanders of the harmonic map flow

TL;DR

This paper proves the existence of weak expanding solutions to the harmonic map flow when the target manifold is embedded as a hypersurface in Euclidean space and the initial map is Lipschitz and homotopic to a point.

Contribution

It establishes the existence of such solutions under specific geometric and initial condition assumptions, extending understanding of harmonic map flow behavior.

Findings

Existence of weak expanding solutions for harmonic map flow.

Solutions exist when target is a hypersurface in Euclidean space.

Initial maps can be Lipschitz and homotopic to a constant.

Abstract

We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded as a hypersurface of some Euclidean space and the initial condition is a Lipschitz map that is homotopic to a constant.