Minimal and constant mean curvature surfaces: their geometric and topological properties.

虽然最小和恒定平均曲率(CMC)曲面的理论是一个纯粹的数学理论，但这种曲面公开地存在于自然界中，并在许多材料科学中进行研究。这使得这一理论更加激动人心。如果我们拿一根封闭的电线，把它浸入肥皂水和肥皂水中，形成的肥皂膜实际上是一个极小的表面，而肥皂膜的物理性质在19世纪50年代就已经由Platform研究过了。肥皂膜两侧的气压是相等且恒定的。然而，如果我们改变一侧的压力，例如通过在上面吹气，我们得到的新表面就是我们所说的肥皂泡。肥皂泡是CMC表面。更准确地说，极小曲面和CMC曲面分别是肥皂膜和肥皂泡的数学理想化。肥皂膜和气泡的平均曲率是一个与膜侧面的压力差成正比的量。对于肥皂膜/最小表面，压差的值是零，因此平均曲率的值是零，而对于肥皂泡/CMC表面，它是非零常量。由于小气泡内部的压力大于大气泡内部的压力，所以小气泡的恒定平均曲率大于大气泡的恒定平均曲率。极小曲面和CMC曲面也具有与面积相关的重要的最小化特性。在跨越给定边界的所有曲面中，肥皂膜/极小曲面是面积局部最小的曲面；肥皂气泡/CMC曲面在体积约束下局部最小化面积。本项目旨在研究极小曲面和CMC曲面的几个关键几何性质。粗略地说，我打算证明关于嵌入到平坦三维流形中的CMC曲面的几个结果，包括当曲面是有界亏格紧的且环境流形是紧的时的面积估计。我还计划研究嵌入到一般三维流形中的极小曲面或CMC曲面序列的极限。

While the theory of minimal and constant mean curvature (CMC) surfaces is a purely mathematical one, such surfaces overtly present themselves in nature and are studied in many material sciences. This makes the theory more exciting. If we take a closed wire and dip it in and out of soapy water, the soap film that forms across the loop is in fact a minimal surface and the physical properties of soap films were already studied by Plateau in the 1850s. The air pressure on the sides of soap films is equal and constant. However, if we change the pressure on one side, for instance by blowing air on it, the new surface that we obtain is what we call a soap bubble. A soap bubble is a CMC surface. More precisely, minimal and CMC surfaces are, respectively, mathematical idealisation of soap films and soap bubbles. The mean curvature of a soap film and bubble is a quantity that is proportional to the pressure difference on the sides of the film. The value of the pressure difference, and therefore of the mean curvature, is zero for a soap film/minimal surface and it is non-zero constant for a soap bubble/CMC surface. Since the pressure inside a small bubble is greater than the pressure inside a big one, the constant mean curvature of a small bubble is greater than the constant mean curvature of a big one. Minimal and CMC surfaces also enjoy crucial minimising properties relative to area. Among all surfaces spanning a given boundary, a soap film/minimal surface is one with locally least area; soap bubbles/CMC surfaces locally minimise area under a volume constraint. This project aims to investigate several key geometric properties of minimal and CMC surfaces. Roughly speaking, I intend to prove several results about CMC surfaces embedded in a flat three-dimensional manifold, including area estimates when the surfaces are compact with bounded genus and the ambient manifold is compact. I also plan to study the limits of a sequence of minimal or CMC surfaces embedded in a general three-dimensional manifold.