Cambridge Journal of Mathematics

Volume 10 (2022)

Number 4

Topological uniqueness for self-expanders of small entropy

Pages: 785 – 833



Jacob Bernstein (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Lu Wang (Department of Mathematics, Yale University, New Haven, Connecticut, U.S.A.)


For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.


self-expander, mean curvature flow, asymptotically conical hypersurfaces, isotopy

2010 Mathematics Subject Classification

Primary 35J20, 35K93, 53A10, 53C44. Secondary 57Q37.

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The first-named author was partially supported by the NSF Grants DMS- 1609340 and DMS-1904674 and the Institute for Advanced Study with funding provided by the Charles Simonyi Endowment.

The second-named author was partially supported by an Alfred P. Sloan Research Fellowship, the NSF Grants DMS-2018220 (formerly DMS-1811144) and DMS-2018221 (formerly DMS-1834824), the Office of the Vice Chancellor for Research and Graduate Education at University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation, a Vilas Early Career Investigator Award, and a von Neumann Fellowship by the Institute for Advanced Study with funding from the Zürich Insurance Company and the NSF Grant DMS-1638352.

Received 21 September 2020

Published 21 October 2022