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arXiv · 2410.06183

Asymptotic circularity of immortal area-preserving curvature flows

Tatsuya Miura
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Abstract

For a class of area-preserving curvature flows of closed planar curves, we prove that every immortal solution becomes asymptotically circular without any additional assumptions on initial data. As a particular corollary, every solution of zero enclosed area blows up in finite time. This settles an open problem posed by Escher--Ito in 2005 for Gage's area-preserving curve shortening flow, and moreover extends it to the surface diffusion flow of arbitrary order. We also establish a general existence theorem for nontrivial immortal solutions under almost circularity and rotational symmetry.

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Tatsuya Miura. 2024-10-08. Asymptotic circularity of immortal area-preserving curvature flows. https://arxiv.org/abs/2410.06183

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