关于最小至最大CMC超表面的孤立奇点和通用规律性
Costante Bellettini1, Kobe Marshall-Stevens2
1Department of Mathematics, University College London, 25 Gordon St, London, WC1H 0AY UK.
Journal of geometric analysis
|March 21, 2025
概括
研究人员证明,由艾伦-卡恩最小-最大过程产生的恒定平均曲率超表面是奇点附近的局部区域最小化器. 这证实了触角是面积最小化的,并保证了在正的里奇曲率多元体中存在这样的超面.
科学领域:
- 不同几何学微分几何学
- 几何分析 几何分析
- 拓学的拓学
背景情况:
- 具有正Ricci曲率的紧的里曼的多元体是几何学的基本对象.
- 艾伦-卡恩最小-最大过程产生恒定平均曲率 (CMC) 的超表面.
- 了解这些超表面的规律性和最小化性质,特别是在奇点附近,至关重要.
研究的目的:
- 证明由艾伦-卡恩最小-最大过程产生的CMC超表面是围绕孤立奇点的区域函数的局部最小化器.
- 为了确定奇点上的触角是面积最小化的.
- 为了证明在具有正Ricci曲率的通用8维紧的里曼纳模组中存在封闭的嵌入式光滑的CMC超表面.
主要方法:
- 通过艾伦-卡恩最小-最大程序生成的CMC超表面的分析.
- 在孤立的奇点周围进行局部规律性分析.
- 在奇点上触角圆的面积最小值.
- 一种手术程序,用于构建封闭的嵌入式光滑的CMC超表面.
主要成果:
- 来自艾伦-卡恩最小-最大程序的每个CMC超表面都是近孤立奇点的区域类型函数的局部最小化器.
- 这些超表面的孤立奇点上的触角是面积最小化的.
- 对于任何实数H,存在一个封闭的嵌入式光滑的CMC超表面与平均曲率H在一个通用的8维紧的里曼的多元体与正的里奇曲率.
结论:
- 该研究确定了通过min-max程序生成的CMC超表面的重要规律性和最小化特性.
- 这些结果有助于理解正曲空间中CMC超表面的存在和几何性质.
- 这项工作将以前对最小超表面的结果扩展到更一般的恒定平均曲率H的情况.
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