A relation between mean curvature flow solitons and minimal submanifolds

We derive a one to one correspondence between conformal solitons of the mea n curvature flow in an ambient space N and minimal submanifolds in a differ ent ambient space (N) over tilde, where (N) over tilde equals R x N equippe d with a warped product metric and show that a submanifold in N converges t o a conformal soliton under the mean curvature flow in N if and only if its associated submanifold in (N) over tilde converges to a minimal submanifol d under a rescaled mean curvature flow in (N) over tilde. We then define a notion of stability for conformal solitons and obtain L-p estimates as well as pointwise estimates for the curvature of stable solitons.