Soliton solutions of the mean curvature flow and minimal hypersurfaces

TL;DR

This paper investigates the relationship between X-pseudosoliton hypersurfaces and minimal hypersurfaces, establishing an equivalence under specific conditions related to the vector field being a gradient, with counterexamples in lower dimensions.

Contribution

It proves that the Monge-Ampère systems for X-pseudosoliton and minimal hypersurfaces are equivalent if and only if X is a gradient vector field, extending understanding of soliton solutions.

Findings

Equivalence holds iff X is a gradient of a function u.

G' is conformally related to g via exp(-2u).

Counterexamples show the equivalence fails for surfaces.

Abstract

Let (M,g) be an oriented Riemannian manifold of dimension at least 3 and X a vector field on M. We show that the Monge-Amp\`ere differential system (M.A.S.) for X-pseudosoliton hypersurfaces on (M,g) is equivalent to the minimal hypersurface M.A.S. on (M,g') for some Riemannian metric g', if and only if X is the gradient of a function u, in which case g'=exp(-2u)g. Counterexamples to this equivalence for surfaces are also given.