Minimal surfaces, constant mean curvature surfaces, and the isoperimetric problem

最小曲面、常平均曲率曲面和等周问题

• 批准号: 5406858
• 负责人: Professor Dr. Uwe Abresch
• 金额: --
• 依托单位: Lehrstuhl Analysis, insbes. Differentialgeometrie und Dynamische Systeme
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2008-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5406858?language=en
• 关键词: Minimal; surfaces; constant; mean; curvature

... We propose to study the isoperimetric problem for symmetric spaces M = G/K of non-compact type. There are three reasons to believe that this is a natural setting: Firstly, symmetric spaces of non-compact type provide enough homogeneity in order to avoid minimizing sequences that disappear at infinity. Secondly, they are Hadamard manifolds, and this may be the basis for ruling out discontinuous changes of the shape of the extremal domains W when the volume v varies. The latter phenomenon is observed e.g. in complex-projective spaces. Finally, the ring of G-invariant differential operators on a symmetric space is commutative, and thus there is a rich supply of spherical functions that obey interesting non-linear differential identities; hence we may hope to obtain explicit descriptions of the isoperimetric domains W.

..。我们建议研究非紧型对称空间M=G/K的等周问题。有三个理由相信这是一个自然的设置：首先，非紧凑型的对称空间提供了足够的同质性，以避免最小化在无穷大处消失的序列。其次，它们是Hadamard流形，这可能是排除当体积v变化时极值区域W的形状不连续变化的基础。后一种现象是在复射影空间中观察到的。最后，对称空间上的G-不变微分算子环是可交换的，因此有丰富的球面函数满足有趣的非线性微分恒等式；因此，我们有望得到等周域W的显式刻画。