Short time existence for harmonic map heat flow with time-dependent metrics

TL;DR

This paper proves short time existence of harmonic map heat flow on manifolds with time-dependent metrics, extending previous results to cases involving Ricci flows and complete noncompact manifolds.

Contribution

It generalizes short time existence results for harmonic map heat flow to settings with evolving metrics, including Ricci flows on complete noncompact manifolds.

Findings

Established short time existence under Ricci flow conditions.

Extended prior results to complete noncompact manifolds.

Provided conditions for uniform equivalence of metrics over time.

Abstract

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.