Short-Time Existence for Harmonic Map Heat Flow with Time-Dependent Metrics

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 (Invent Math 105(1):1–46, 1991) and Chen-Zhu (J Differ Geom 74:119–154, 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, under the assumption: (i) | Rm (g(t)) | ≤ a/ t; (ii) g(t) is uniformly equivalent to g(0); and (iii) the initial map is smooth and with uniformly bounded energy density.