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，但这种曲面在自然界中公开存在，并在许多材料科学中进行研究。这使得理论更加令人兴奋。如果我们拿一根闭合的金属丝，把它浸在肥皂水里，然后从肥皂水里出来，在环上形成的肥皂膜实际上是一个最小的表面，19世纪50年代，Plateau已经研究了肥皂膜的物理特性。肥皂膜两侧的空气压力相等且恒定。然而，如果我们改变一侧的压力，例如通过吹空气，我们获得的新表面就是我们所说的肥皂泡。肥皂泡是CMC表面。更确切地说，最小和CMC表面分别是肥皂膜和肥皂泡的数学理想化。肥皂膜和肥皂泡的平均曲率是与膜两侧的压力差成比例的量。对于肥皂膜/最小表面，压力差的值以及因此平均曲率的值为零，并且对于肥皂泡/CMC表面，压力差的值为非零常数。由于小气泡内的压力大于大气泡内的压力，因此小气泡的恒定平均曲率大于大气泡的恒定平均曲率。最小和CMC表面还具有相对于面积的关键最小化特性。在所有跨越给定边界的表面中，肥皂膜/最小表面是具有局部最小面积的表面;肥皂泡/CMC表面在体积约束下局部最小化面积。这个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，我打算证明几个结果的CMC曲面嵌入在一个平坦的三维流形，包括面积估计时，表面是紧的有界亏格和周围的流形是紧的。我还计划研究嵌入在一般三维流形中的一系列极小或CMC曲面的极限。

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.