TY - JOUR

T1 - A note on the selfsimilarity of limit flows

AU - Choi, Beomjun

AU - Haslhofer, Robert

AU - Hershkovits, Or

N1 - Publisher Copyright:
© 2020 by the authors

PY - 2021/3

Y1 - 2021/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85100671763&partnerID=8YFLogxK

U2 - 10.1090/proc/15251

DO - 10.1090/proc/15251

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AN - SCOPUS:85100671763

SN - 0002-9939

VL - 149

SP - 1239

EP - 1245

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -