8 维流形中奇异等周区域的存在性

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-20 DOI:10.1007/s00526-024-02748-y

Gongping Niu

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引用次数: 0

摘要

众所周知，光滑紧凑（(n+1)\）-流形中的等周区域本身是光滑的，直到一个至多 8 维的闭集。在本注释中，我们构造了一个 8 维紧凑光滑流形，其唯一的等周区域具有流形一半的体积，表现出两个孤立奇点。这与流形是空间形式的情况截然不同，后者的等周区域在每一维都是光滑的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

Existence of singular isoperimetric regions in 8-dimensional manifolds

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Existence of singular isoperimetric regions in 8-dimensional manifolds

It is well known that isoperimetric regions in a smooth compact \((n+1)\)-manifold are themselves smooth, up to a closed set of codimension at most 8. In this note, we construct an 8-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. This stands in contrast with the situation in which a manifold is a space form, where isoperimetric regions are smooth in every dimension.

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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.