A Relation between Mean Curvature Flow Solitons and Minimal Submanifolds

Abstract

We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space where equals ℝ × N equipped with a warped product metric and show that a submanifold inN converges to a conformal soliton under the mean curvature flow in N if and only if its associatedsubmanifold in converges to a minimal submanifold under a rescaled mean curvature flow in. We then define a notion of stability for conformal solitons and obtain Lp estimates as well as pointwise estimates for the curvature of stable solitons.