Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-08-05 DOI:10.1007/s00526-024-02792-8

Jonas Hirsch, Rob Kusner, Elena Mäder-Baumdicker

引用次数: 0

Abstract

We study complete minimal surfaces in \(\mathbb {R}^n\) with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy \(\mathcal {W}: =\frac{1}{4} \int |\vec H|^2\). In codimension one, we prove that the \(\mathcal {W}\)-Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly \(m-3=\frac{\mathcal {W}}{4\pi }-3\). We also consider several geometric properties—for example, the property that all m asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the \(\mathcal {W}\)-Morse index of their inverted surfaces.

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无穷远处完全极小曲面的几何及其反转的威尔莫尔指数

我们研究的是\(\mathbb {R}^n\)中具有有限总曲率和内嵌平面末端的完整极小曲面。在通过反转进行保角压实之后，这些曲面产生了静止于威尔莫尔弯曲能 \(\mathcal {W}: =\frac{1}{4} \int |\vec H|^2\) 的例子。在标度为一的情况下，我们证明任何倒置的极小球面或实投影面的莫尔斯指数（\(\mathcal {W}\)-Morse index）正好是\(m-3=\frac{mathcal {W}{4\pi }-3\)。我们还考虑了这些极小曲面的几个几何性质--例如，所有 m 个渐近平面在一个点相遇的性质，并探讨了它们与倒转曲面的 \(\mathcal {W}\)-Morse 索引的关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.