Rigidity of generic singularities of mean curvature flow

By T. Colding, T. Ilmanen, W. Minicozzi. Publications mathématiques (Bures-sur-Yvette), 2013

89 citations recorded by the source corpus.

Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, Colding and Minicozzi II (Ann. Math. 175(2):755–833, 2012) showed that the only generic are round cylinders Sk×Rn−k. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder. To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas developed here have applications to other flows. Our results hold in all dimensions and do not require any a priori smoothness.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6fbf83751453189527204cf58b7c03d9d89b2bc7

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. W. Minicozzi
3. T. Colding
4. Kyeongsu Choi
5. Or Hershkovits
6. B. White
7. L. Alías
8. M. Rigoli

In Geometry and Topology

1. Robert Haslhofer
2. W. Minicozzi
3. T. Colding
4. Kyeongsu Choi
5. Or Hershkovits
6. B. White
7. L. Alías
8. J. D. de Lira

In Mathematical Physics

1. T. Colding
2. W. Minicozzi
3. Robert Haslhofer
4. Or Hershkovits
5. J. D. de Lira
6. Kyeongsu Choi
7. B. White
8. M. Rigoli

Citations in the corpus

• Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields (2017, background): “One of the main research topics on mean curvature flow is the study of self-similar solutions particularly as models of singularities after suitable rescalings of the flow [27], [28], [50], [15], [16], [17].”
• Ancient asymptotically cylindrical flows and applications (2019, background): “This shrinker can not be the cylinder, as the cylinder is isolated in the space of shrinkers by [CIM15].”
• The singular set of mean curvature flow with generic singularities (2014, background): “If at least one blowup is cylindrical, then all are by [9] and the axis of the cylinder is unique by [12].”