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Mean curvature flow with generic initial data

https://doi.org/10.1007/s00222-024-01258-0

Chodosh, Otis; Choi, Kyeongsu; Mantoulidis, Christos; Schulze, Felix (April 2024, Inventiones mathematicae)

Abstract We show that the mean curvature flow of generic closed surfaces in$$\mathbb{R}^{3}$$ $R^{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in$$\mathbb{R}^{4}$$ $R^{4}$ is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.
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Mean curvature flow with generic low-entropy initial data

https://doi.org/10.1215/00127094-2023-0034

Chodosh, Otis; Choi, Kyeongsu; Mantoulidis, Christos; Schulze, Felix (January 2024, Duke Mathematical Journal)

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Convergence of Gauss curvature flows to translating solitons

https://doi.org/10.1016/j.aim.2022.108207

Choi, Beomjun; Choi, Kyeongsu; Daskalopoulos, Panagiota (March 2022, Advances in Mathematics)

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Ancient gradient flows of elliptic functionals and Morse index

https://doi.org/10.1353/ajm.2022.0010

Choi, Kyeongsu; Mantoulidis, Christos (January 2022, American Journal of Mathematics)

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Convergence of curve shortening flow to translating soliton

https://doi.org/10.1353/ajm.2021.0032

Choi, Beomjun; Choi, Kyeongsu; Daskalopoulos, Panagiota (August 2021, American journal of mathematics)

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Uniqueness of convex ancient solutions to mean curvature flow in R^3.

https://doi.org/10.1007/s00222-019-00859-4

Brendle, Simon; Choi, Kyeongsu (July 2019, Inventiones mathematicae)

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are κ-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in R^3, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in R^3 which is strictly convex and noncollapsed.
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