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 smooth family of complete metrics in the domain manifold. Our results generalize by Li-Tam (Invent Math 105(1):1–46, 1991) and Chen-Zhu (J Differ Geom 74:119–154, 2006). particular, prove along Ricci g(t) on M into manifold curvature bounded from above, under assumption: (i) $$|\text {Rm}(g(t))|\le a/t$$ ; (ii) is uniformly equivalent to g(0); (iii) initial energy density.