Communications in Analysis and Geometry

Volume 31 (2023)

Number 1

Avoidance for set-theoretic solutions of mean-curvature-type flows

Pages: 31 – 67



Or Hershkovits (Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, Israel)

Brian White (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


We give a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint provided one of the subsolutions is compact; previously, this was only known for Euclidean space (with no ambient vectorfield). We also give a simple proof of a version of Ilmanen’s interpolation theorem.

The first author was partially supported by an AMS-Simons Travel Grant.

The second author was partially supported by NSF grants DMS-1404282 and DMS-1711293.

Received 12 September 2018

Accepted 7 August 2020

Published 21 September 2023