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 N , where N equals IR × N equipped with a warped product metric and show that a submanifold in N converges to a conformal soliton under the mean curvature flow in N if and only if its associated submanifold in N converges to a minimal submanifold under a rescaled mean curvature flow in N . 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.

Stability of translating solutions to mean curvature flow

“…Stability of non-compact gradient Kähler-Ricci solitons is investigated in [5]. Stability of the grim reaper, a translating curve solving mean curvature flow, is considered in [13]. Richard Hamilton [10] mentions non-convex, complete translating solutions to mean curvature flow.…”

Conformal solitons to the mean curvature flow and minimal submanifolds

“…As an application of the general picture just described, we can then give affirmative answers to two natural questions proposed by Smoczyk (11). Namely, we have…”

“…The mean curvature flow of M in N is a family of immersions which satisfies where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textbf {H}_t$\end{document} denotes the mean curvature vector field of F t ( M )⊂ N . Following Smoczyk (11), given a conformal vector field \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textbf {X}$\end{document} on N , we say that M is a conformal soliton to the mean curvature flow if M satisfies the following equation where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textbf {H}$\end{document} is the mean curvature vector of M in N and ⟂ denotes the projection onto the normal bundle. Recall that a vector field \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textbf {X}$\end{document} is conformal if in local coordinate, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{\nabla }_j X_i+\overline{\nabla }_i X_j=2\lambda g_{ij}$\end{document} for some smooth function λ.…”

“…Recall that a vector field \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textbf {X}$\end{document} is conformal if in local coordinate, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{\nabla }_j X_i+\overline{\nabla }_i X_j=2\lambda g_{ij}$\end{document} for some smooth function λ. As in (11), we will consider the special conformal vector fields \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textbf {X}$\end{document} which satisfy for some smooth function λ. Here, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{\nabla }$\end{document} is the Levi‐Civita connection on N .…”

“…Smoczyk (11) found a one to one correspondence between conformal solitons satisfying (1.4) with codimension one in an ambient Riemannian manifold ( N , g ) and minimal submanifolds in the manifold \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\tilde{N} = N \times \textbf {R}$\end{document} with a warped product metric \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\tilde{g}$\end{document}. (See Section 3 for more details.)…”