A note on the selfsimilarity of limit flows
| Autor: | Beomjun Choi, Robert Haslhofer, Or Hershkovits |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Differential Geometry
Sequence Mean curvature flow Partial differential equation 010308 nuclear & particles physics Applied Mathematics General Mathematics Open problem 010102 general mathematics Mathematical analysis Regular polygon 01 natural sciences Physics::Fluid Dynamics Mathematics - Analysis of PDEs Singularity Differential Geometry (math.DG) 0103 physical sciences FOS: Mathematics Gravitational singularity Limit (mathematics) 0101 mathematics Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Proceedings of the American Mathematical Society. 149:1239-1245 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/proc/15251 |
| Popis: | It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establish a local version of this equivalence for neck singularities in arbitrary dimension. In particular, we see that the ancient ovals occur as limit flows if and only if there is a sequence of spherical singularities converging to a neck singularity. Comment: 5 pages |
| Databáze: | OpenAIRE |
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