Exponential convergence rate of the harmonic heat flow

TL;DR

This paper proves that harmonic heat flow converges exponentially fast to a non-degenerate harmonic map under certain geometric conditions, providing insights into the flow's convergence behavior.

Contribution

It establishes exponential convergence rates for harmonic heat flow into non-positively curved manifolds when converging to a non-degenerate harmonic map.

Findings

Convergence rate is exponential in the L^2 norm.

Harmonic heat flow converges to non-degenerate harmonic maps.

Results depend on the non-positive curvature condition.

Abstract

We consider the harmonic heat flow for maps from a compact Riemannian manifold into a Riemannian manifold that is complete and of non-positive curvature. We prove that if the harmonic heat flow converges to a limiting harmonic map that is a non-degenerate critical point of the energy functional, then the rate of convergence is exponential (in the $L^2$ norm).