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Stable hypersurfaces with prescribed mean curvature

具有规定平均曲率的稳定超曲面

基本信息

批准号：
EP/S005641/1
负责人：
Costante Bellettini
金额：
$ 36.58万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS005641%2F1
关键词：
Stable hypersurfaces prescribed mean curvature

项目摘要

The area of a surface governs many physical phenomena. Nature tends to optimise shapes by finding equilibrium positions dictated by a minimality property- roughly speaking, it prefers to use as little area as possible. Well-known examples of this phenomenon are soap films. As early as the mid 19th century, the physicist Plateau conducted experiments in which he immersed a closed wire in and out of a soap solution. The resulting soap film is a minimal surface, i.e. it locally minimizes area among surfaces spanning the given wire (it avoids wasting soap). Of particular interest are configurations of ``stable'' equilibrium, i.e. under any slight perturbation the film will go back to its initial position. Similarly, in the case of soap bubbles, it is again a minimality property of area that dictates their shape (e.g. spherical bubbles), with the difference that this time the minimality is achieved under the constraint of a fixed enclosed volume (how much air the bubble contains): the surface obtained is characterized by having constant mean curvature (CMC). The mean curvature of a soap film or bubble is a geometric quantity that is proportional to the pressure difference on the sides of the film. The optimising behaviour observed in these examples is ubiquitous in nature (for example, bees use hexagonal cells because this requires the minimal amount of wax for tiling a planar portion); the following is a further example, taken from capillarity theory, and it is very relevant to the present project. Consider a stable equilibrium configuration for a liquid that is surrounded by air, subject to surface tension and to the action of external body forces, such as gravitational energy. By a principle of energy optimization, the equilibrium configuration is once again dictated by a partial differential equation whose geometric content is to prescribe the mean curvature of the interface (the surface that separates liquid and air). More precisely, in the absence of gravity or other external forces, the condition is that the mean curvature is constant (CMC surfaces); in the presence of a non-zero potential, for example, a gravitational one, the mean curvature is prescribed up to an additive constant by the value of the potential. Modern geometry is not limited to surfaces in three-dimensional space and this has allowed, and will for time to come, far-reaching applications, from relativity theory and black holes to engineering. It is therefore natural to introduce hypersurfaces (a generalization to arbitrary dimensions of a surface in three-dimensional space) of dimension n that sit in an ambient space of dimension n+1. In mathematics this ambient space is a Riemannian manifold, i.e. a space with compatible notions of length and angle that permit the computation of area, volume, etc.In this project I study stable hypersurfaces whose mean curvature is prescribed by a given function on the ambient Riemannian manifold (special cases of which include minimal and constant-mean-curvature hypersurfaces). The project aims to address the fundamental geometric question of existence of closed hypersurfaces of this type in arbitrary closed Riemannian manifolds, employing an analytic framework (regularity and compactness results) that I recently developed. The successful completion of this project will be a pathway towards a more complete understanding of interfaces between liquids and air (as in the capillarity model above).

曲面的面积决定着许多物理现象。大自然倾向于通过寻找由极小属性决定的平衡位置来优化形状-粗略地说，它喜欢使用尽可能小的面积。这种现象的著名例子是肥皂膜。早在世纪中期，物理学家Plateau就进行了一项实验，他将一根封闭的电线浸入肥皂溶液中。所得到的肥皂膜是最小表面，即，它局部地最小化跨越给定线的表面之间的面积（它避免浪费肥皂）。特别令人感兴趣的是“稳定”平衡的构型，即在任何轻微的扰动下，膜将回到其初始位置。类似地，在肥皂泡的情况下，同样是面积的最小性特性决定了它们的形状（例如球形气泡），不同之处在于，这次在固定封闭体积（气泡包含多少空气）的约束下实现最小性：所获得的表面的特征在于具有恒定的平均曲率（CMC）。肥皂膜或肥皂泡的平均曲率是与膜两侧的压力差成比例的几何量。在这些例子中观察到的优化行为在自然界中是普遍存在的（例如，蜜蜂使用六边形细胞，因为这需要最少量的蜡来平铺平面部分）;下面是另一个例子，来自毛细作用理论，它与本项目非常相关。考虑一个被空气包围的液体的稳定平衡构型，受到表面张力和外部体力的作用，如重力能。根据能量优化原理，平衡构型再次由偏微分方程决定，该偏微分方程的几何内容是规定界面（分离液体和空气的表面）的平均曲率。更确切地说，在没有重力或其他外力的情况下，条件是平均曲率是常数（CMC曲面）;在存在非零势（例如重力势）的情况下，平均曲率由势的值规定为加性常数。现代几何学并不局限于三维空间中的表面，这使得从相对论、黑洞到工程学的广泛应用成为可能。因此很自然地引入了n维的超曲面（三维空间中曲面的任意维数的推广），它们位于n+1维的环境空间中。在数学中，这个周围的空间是一个黎曼流形，即一个空间与兼容的概念的长度和角度，允许计算面积，体积等。在这个项目中，我研究稳定的超曲面，其平均曲率是由一个给定的函数规定的周围黎曼流形（特殊情况下，其中包括最小和常数平均曲率超曲面）。该项目的目的是解决基本的几何问题存在的这种类型的封闭超曲面在任意封闭黎曼流形，采用分析框架（正则性和紧性结果），我最近开发的。该项目的成功完成将是一条通往更完整地理解液体和空气之间界面的途径（如上述毛细作用模型）。