Conformal solitons to the mean curvature flow and minimal submanifolds

Abstract

In this note we prove a correspondence, first found by Smoczyk in the hypersurface case, between conformal solitons to the mean curvature flow in an ambient manifold N and minimal submanifolds in a different space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$N\times \textbf {R}$\end{document}. This naturally leads to a new natural stability notion for conformal solitons. We show that this corresponds to the recent Colding‐Minicozzi's F‐stability for codimension one self‐shrinkers in euclidean space. In this spirit we can give some classification results for stable conformal solitons, answering two questions of Smoczyk.

Invariant Translators of the Heisenberg Group

“…Roughly speaking this property holds if the coefficients of all the terms in v 2 vanish. For example if we take v 2 = π 2 m with m ∈ Z, then (6.1) must not depend on m. Since a 2 1 + a 2 2 = 0 and c = 0 we derive that it is possible if and only if γ 1 = γ 2 = 0, having a contradiction. Remark 6.1 Theorem 1.3, part (4) says in particular that in Nil 3 the only helicoidal surfaces that are translators are those with vertical axis, i.e., when a = b = 0.…”

“…A unit normal vector field is has to be independent on v 2 , but this is impossible if a = 0 because,in this case, this property holds if only if γ 1 = γ 2 = 0. The proof in a consequence of the fact that the functions 1, cos 2 2 cos(v 2 ) are linearly independent.…”

“…The first one is due to Ilmanen [17]: M is a translator for the Euclidean metricḡ if and only if M is minimal with respect to the conformal metricg p = e − 2 nḡ ( p,V )ḡ p . The second one has been discovered by Smoczyk [26] in case of hypersurfaces and generalized by Arezzo and Sun [2] for higher codimensional submanifolds: it gives a correspondence between translators in the Euclidean space R n and minimal submanifolds in R n × R equipped with a suitable warped product metric. The two approaches are very useful because they allow to use tools from the minimal surface theory in the study of translators.…”

Rigidity results and topology at infinity of translating solitons of the mean curvature flow

“…Moreover, it has been proved by C. Arezzo and J. Sun, [3], but also by L. Ma, using different techniques, [27], that the grim reaper is a f -stable translator in R 2 . Similar computations show that also the grim reaper cylinders are f -stable.…”

Hamiltonian L-stability of Lagrangian translating solitons

“…With the above known results for self-shrinkers, it is natural to think that translating solitons might also have some similar properties. In fact, translating solitons are also critical points for an F -functional, which was studied by some people (See [3], [11] and [13] for example ). The F -functional is defined by…”