- Title
- Two-sided non-collapsing curvature flows
- Creator
- Andrews, Ben; Langford, Mat
- Relation
- Annali della Scuola Normale Superiore di Pisa, Classe di Scienze Vol. XV, Issue Special, p. 543-560
- Publisher Link
- http://dx.doi.org/10.2422/2036-2145.201310_001
- Publisher
- Scuola Normale Superiore di Pisa
- Resource Type
- journal article
- Date
- 2016
- Description
- It was recently shown that embedded solutions of curvature flows in Euclidean space with concave (convex), degree one homogeneous speeds are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the first part of the paper, we show that locally convex solutions are exterior non-collapsing for a larger class of speed functions than previously considered; more precisely, we show that the previous results hold when convexity of the speed function is relaxed to inverse-concavity. We note that inverse-concavity is satisfied by a large class of concave speed functions. Thus, as a consequence, we obtain a large class of two-sided non-collapsing flows, whereas previously two-sided non-collapsing was only known for the mean curvature flow. In Section 3, we demonstrate the utility of two sided non-collapsing with a straightforward proof of convergence of compact, convex hypersurfaces to round points. The proof of the non-collapsing estimate is similar to those of the previous results mentioned, in that we show that the exterior ball curvature is a viscosity supersolution of the linearised flow equation. The new ingredient is the following observation: Since the function which provides an upper support in the derivation of the viscosity inequality is defined on M x M (or T M in the ‘boundary case’), whereas the exterior ball curvature and the linearised flow equation depend only on the first factor, we are privileged with a freedom of choice in which second derivatives from the extra directions to include in the calculation. The optimal choice is closely related to the class of inverse-concave speed functions.
- Subject
- curvature flows; Euclidean space; convex; non-collapsing; speed functions
- Identifier
- http://hdl.handle.net/1959.13/1442242
- Identifier
- uon:41634
- Identifier
- ISSN:0391-173X
- Language
- eng
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