The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3348–3372 | Cite as

A Finiteness Theorem Via the Mean Curvature Flow with Surgery

  • Alexander MramorEmail author


In this article, we use the recently developed mean curvature flow with surgery for 2-convex hypersurfaces to prove certain isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger’s finiteness theorem) for the space of 2-convex closed embedded hypersurfaces in \({\mathbb {R}}^{n+1}\).


Mean curvature flow Surgery Mean convex Two-convex 

Mathematics Subject Classification




The author wishes to thank his advisor Rick Schoen, in addition to his great patience and generosity, for mentioning the Smale conjecture and suggesting flow methods might eventually be used to reprove it, which inspired this work (although the end product falls far short of the goal). The author also thanks Hung Tran and Dave Wiygul for several stimulating conversations throughout the course of this work.


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© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California IrvineIrvineUSA

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