Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

Abstract

In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant $\alpha =-1$ (or $+1$), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.