Stable hypersurfaces with prescribed mean curvature

表面的面积支配着许多物理现象。大自然倾向于通过寻找由极小性属性决定的平衡位置来优化形状--粗略地说，它倾向于使用尽可能小的面积。这种现象的著名例子是肥皂片。早在19世纪中叶，物理学家普兰特就进行了一项实验，在实验中，他将一根封闭的电线浸入肥皂溶液中，然后从肥皂溶液中取出。生成的肥皂膜是一个最小曲面，即它局部最小化了跨越给定导线的曲面之间的面积(它避免了肥皂的浪费)。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，薄膜将返回到其初始位置。同样，在肥皂泡的情况下，决定其形状的也是面积的极小属性(例如球形气泡)，不同的是这次最小值是在固定封闭体积(气泡包含多少空气)的约束下实现的：所获得的表面的特征是具有恒定的平均曲率(CMC)。肥皂膜或气泡的平均曲率是一个几何量，它与膜侧面的压力差成正比。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，蜜蜂使用六角形蜂窝，因为这需要最少的蜡来平铺平面部分)；以下是取自毛细管理论的另一个例子，它与本项目非常相关。考虑一种稳定的平衡构型，液体被空气包围，受到表面张力和外部体力的作用，如引力能。根据能量优化原理，平衡构型再次由偏微分方程式决定，该偏微分方程式的几何内容是规定界面(分隔液体和空气的表面)的平均曲率。更准确地说，在没有重力或其他外力的情况下，条件是平均曲率是恒定的(CMC曲面)；在存在非零势的情况下，例如引力势，平均曲率由势值规定为一个附加常数。现代几何学并不局限于三维空间中的表面，这使得从相对论到黑洞到工程的深远应用得以实现，并将在未来一段时间内实现。因此，在n+1维的环境空间中引入n维超曲面(三维空间中曲面的任意维的推广)是很自然的。在数学上，这个环境空间是黎曼流形，即具有相容的长度和角度概念的空间，允许计算面积、体积等。在这个项目中，我研究稳定的超曲面，其平均曲率由环境黎曼流形上的给定函数规定(其特例包括最小和常平均曲率的超曲面)。该项目旨在利用我最近发展的分析框架(正则性和紧致性结果)来解决任意闭黎曼流形中这类闭超曲面存在的基本几何问题。这一项目的成功完成将有助于更全面地了解液体和空气之间的界面(如上面的毛细作用模型)。

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).