## A note on the selfsimilarity of limit flows

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- by Beomjun Choi, Robert Haslhofer and Or Hershkovits PDF
- Proc. Amer. Math. Soc.
**149**(2021), 1239-1245

## Abstract:

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.## References

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## Additional Information

**Beomjun Choi**- Affiliation: Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario M5S 2E4, Canada
- MR Author ID: 1269631
- ORCID: 0000-0003-3842-4135
**Robert Haslhofer**- Affiliation: Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario M5S 2E4, Canada
- MR Author ID: 949022
**Or Hershkovits**- Affiliation: Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904, Israel
- MR Author ID: 1168559
- Received by editor(s): October 11, 2019
- Received by editor(s) in revised form: June 28, 2020
- Published electronically: December 31, 2020
- Additional Notes: The second author was partially supported by an NSERC Discovery Grant (RGPIN-2016-04331) and a Sloan Research Fellowship.

The third author was partially supported by a Koret Foundation early career scholar award. - Communicated by: Jiaping Wang
- © Copyright 2020 by the authors
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 1239-1245 - MSC (2020): Primary 53E10
- DOI: https://doi.org/10.1090/proc/15251
- MathSciNet review: 4211877