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Classification of bubble-sheet ovals in $\mathbb{R}^{4}$

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arxiv 2209.04931 v3 pith:MI3ZUFO5 submitted 2022-09-11 math.DG math.AP

Beomjun Choi , Panagiota Daskalopoulos , Wenkui Du , Robert Haslhofer , Natasa Sesum This is my paper · ORCID
classification math.DGmath.AP
keywords mathbbtextrmancientauthorbubble-sheetclassificationconstructedfourth
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abstract

In this paper, we prove that any bubble-sheet oval for the mean curvature flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.

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  1. Passing through nondegenerate singularities in mean curvature flows

    math.DG 2025-01 conditional novelty 7.0 of 10

    Mean curvature flows through nondegenerate cylindrical singularities undergo an isolated, graphical surgery event whose topology change equals an (n-k)-surgery, matching Morse level sets.

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