Uniqueness Problem for Closed Non-smooth Hypersurfaces with Constant Anisotropic Mean Curvature and Self-similar Solutions of Anisotropic Mean Curvature Flow

研究成果: Chapter in Book/Report/Conference proceedingConference contribution

抄録

An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered surface, and it is a generalization of surface area. Equilibrium surfaces with volume constraint are called CAMC (constant anisotropic mean curvature) surfaces and they are not smooth in general. We show that, if the energy density function is two times continuously differentiable and convex, then, like isotropic (constant mean curvature) case, the uniqueness for closed stable CAMC surfaces holds under the assumption of the integrability of the anisotropic principal curvatures. Moreover, we show that, unlike the isotropic case, uniqueness of closed embedded CAMC surfaces with genus zero in the three-dimensional euclidean space does not hold in general. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. These results are generalized to hypersurfaces in the Euclidean space with general dimension. This article is an announcement of two forthcoming papers by the author.

本文言語英語
ホスト出版物のタイトルMinimal Surfaces
ホスト出版物のサブタイトルIntegrable Systems and Visualisation - Workshops, 2016-19
編集者Tim Hoffmann, Martin Kilian, Katrin Leschke, Francisco Martin
出版社Springer
ページ169-185
ページ数17
ISBN(印刷版)9783030685409
DOI
出版ステータス出版済み - 2021
イベントWorkshop Series of Minimal Surfaces: Integrable Systems and Visualisation, 2016-19 - Cork, アイルランド
継続期間: 3 27 20173 29 2017

出版物シリーズ

名前Springer Proceedings in Mathematics and Statistics
349
ISSN(印刷版)2194-1009
ISSN(電子版)2194-1017

会議

会議Workshop Series of Minimal Surfaces: Integrable Systems and Visualisation, 2016-19
国/地域アイルランド
CityCork
Period3/27/173/29/17

All Science Journal Classification (ASJC) codes

  • 数学 (全般)

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