TY - JOUR
T1 - Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type
AU - Hershkovits, Or
AU - White, Brian
N1 - Publisher Copyright:
© 2019 Wiley Periodicals, Inc.
PY - 2020/3/1
Y1 - 2020/3/1
N2 - We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝn + 1 remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1, and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow.
AB - We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝn + 1 remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1, and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow.
UR - http://www.scopus.com/inward/record.url?scp=85068607606&partnerID=8YFLogxK
U2 - 10.1002/cpa.21852
DO - 10.1002/cpa.21852
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AN - SCOPUS:85068607606
SN - 0010-3640
VL - 73
SP - 558
EP - 580
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
IS - 3
ER -