Self-Expanders of the Mean Curvature Flow

authored by

Knut Smoczyk

Abstract

We study self-expanding solutions Mm ⊂ Rn of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of selfexpanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a localmaximum, where A denotes the second fundamental form and H the mean curvature vectorof M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ |2/|H|2, where Aξ is the second fundamental form with respect to ξ . As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex selfexpander that is asymptotic to a cone must be strictly convex.

Organisation(s)

Institut für Differentialgeometrie

Type

Artikel

Journal

Vietnam Journal of Mathematics

Volume

49

Pages

433-445

No. of pages

13

Publication date

06.2021

Publication status

Veröffentlicht

Peer reviewed

Yes

ASJC Scopus Sachgebiete

Mathematik (insg.)

Electronic version(s)

https://doi.org/10.1007/s10013-020-00469-1 (Access: Offen)