TY - GEN

T1 - Uniqueness Problem for Closed Non-smooth Hypersurfaces with Constant Anisotropic Mean Curvature and Self-similar Solutions of Anisotropic Mean Curvature Flow

AU - Koiso, Miyuki

N1 - Funding Information:
Acknowledgements The author would like to thank the referee for the valuable comments which helped to improve the manuscript. This work was partially supported by JSPS KAKENHI Grant Number JP18H04487.
Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-030-68541-6_10

DO - 10.1007/978-3-030-68541-6_10

M3 - Conference contribution

AN - SCOPUS:85111122129

SN - 9783030685409

T3 - Springer Proceedings in Mathematics and Statistics

SP - 169

EP - 185

BT - Minimal Surfaces

A2 - Hoffmann, Tim

A2 - Kilian, Martin

A2 - Leschke, Katrin

A2 - Martin, Francisco

PB - Springer

T2 - Workshop Series of Minimal Surfaces: Integrable Systems and Visualisation, 2016-19

Y2 - 27 March 2017 through 29 March 2017

ER -