Embedded self-similar shrinkers of genus 0

By S. Brendle. 2014

137 citations recorded by the source corpus.

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution to the mean curvature flow with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/45e0e2f1d66624a7592c20ca452da0c1f9bf278f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Robert Haslhofer
2. Or Hershkovits
3. Kyeongsu Choi
4. B. White
5. Bruce Kleiner
6. G. Huisken
7. S. Brendle
8. Carlo Sinestrari

In Geometry and Topology

1. Robert Haslhofer
2. Kyeongsu Choi
3. Or Hershkovits
4. B. White
5. Bruce Kleiner
6. S. Brendle
7. G. Huisken
8. A. Neves

In Mathematical Physics

1. Robert Haslhofer
2. Or Hershkovits
3. Kyeongsu Choi
4. B. White

Citations in the corpus

• Ancient asymptotically cylindrical flows and applications (2019, background): “n p 2n . More recently, without any curvature assumptions, Brendle showed that every smooth, two-dimensional shrinker of genus 0 in R3 is either the round sphere, the round cylinder or a flat plane [Bre16]. Those are by no means the only smooth, two-dimensional shrinkers in R3, as examples construct”