The geometry of a surface embedded in a 3-manifold with constant mean curvature

虽然恒定平均曲率曲面理论是一个纯粹的数学理论，但这样的曲面在自然界中明显存在，并在许多材料科学中得到研究。这使得理论更令人兴奋，并为项目增加了价值。如果我们拿一根闭合的电线，把它浸在肥皂水里，在铁环上形成的肥皂膜实际上是一个最小（常数平均曲率等于零）表面，而肥皂膜的物理性质早在19世纪50年代就已经被Plateau研究过了。在跨越给定边界的所有表面中，肥皂膜是面积最小的表面。虽然肥皂膜两侧的气压是相等且恒定的，但如果我们改变一侧的气压，例如向其吹气，我们得到的新表面就是一个肥皂泡。从数学上讲，肥皂泡是平均曲率不变的表面，它们在体积限制下使面积最小化。本项目旨在描述最小和常平均曲率曲面的几个几何性质。粗略地说，我打算证明满足某些弱几何条件的常平均曲率曲面不会过度弯曲。我打算构造一些例子来说明前面的结果在某种意义上是尖锐的。最后，我打算研究某些极小曲面序列的极限。

While the theory of constant mean curvature 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 and adds value to the project. If we take a closed wire and dip it in and out of soapy water, the soap film that forms across the hoop is in fact a minimal (constant mean curvature equal to zero) surface, and the physical properties of soap films were already studied by Plateau in the 1850s. Among all surfaces spanning a given boundary, a soap film is the one with least area. While the air pressure on the sides of a soap film is equal and constant, if we change the pressure on one side, for instance by blowing air on it, the new surface that we obtain is a soap bubble. Mathematically, soap bubbles are constant mean curvature surfaces, and they minimize area under a volume constraint.This project aims to describe several geometric properties of minimal and constant mean curvature surfaces. Roughly speaking, I intend to prove that constant mean curvature surfaces which satisfy certain weak geometric conditions cannot bend too much. I intend to construct examples to show that the previous results are, in a sense, sharp. Finally, I intend to study the limits of certain sequences of minimal surfaces.