## Journal of Differential Geometry

### Convex ancient solutions of the mean curvature flow

#### Abstract

We study solutions of the mean curvature flow which are defined for all negative times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter, or a reverse isoperimetric inequality. We also study the behaviour of uniformly $k$-convex solutions, and consider generalizations to ancient solutions immersed in a sphere.

#### Article information

Source
J. Differential Geom., Volume 101, Number 2 (2015), 267-287.

Dates
First available in Project Euclid: 16 September 2015

https://projecteuclid.org/euclid.jdg/1442364652

Digital Object Identifier
doi:10.4310/jdg/1442364652

Mathematical Reviews number (MathSciNet)
MR3399098

Zentralblatt MATH identifier
1332.53085

#### Citation

Huisken, Gerhard; Sinestrari, Carlo. Convex ancient solutions of the mean curvature flow. J. Differential Geom. 101 (2015), no. 2, 267--287. doi:10.4310/jdg/1442364652. https://projecteuclid.org/euclid.jdg/1442364652