Existence of expanding harmonic map flows to hemispheres

TL;DR

This paper proves the existence of non-trivial self-expanding harmonic map flows from certain initial maps to hemispheres, providing infinitely many solutions and answering a question posed by Struwe.

Contribution

It demonstrates the existence of multiple weak solutions to harmonic map flow starting from non-energy-minimizing 0-homogeneous maps to hemispheres, expanding understanding of harmonic map dynamics.

Findings

Existence of non-trivial self-expanding harmonic map flows.

Construction of infinitely many solutions from a given initial map.

Solutions satisfy the parabolic monotonicity formula.

Abstract

We show the existence of non-trivial self-expanding harmonic map flows starting from non-energy-minimizing 0-homogeneous maps to a regular ball or a closed hemisphere. In particular, given a non-minimizing but stationary 0-homogeneous harmonic map $u_0$ to a closed hemisphere, we construct infinitely many different weak solutions to harmonic map flow starting from $u_0$, all of which satisfy the parabolic monotonicity formula. This answers a question of Struwe.