The shape of nonzero constant mean curvature surfaces embedded in Euclidean space.

这个项目包括一个深入的研究几何非零常数平均曲率曲面嵌入在欧几里德空间。这个项目的主要目标之一是获得一些新的和显着的曲率估计嵌入在欧氏空间的非零常数平均曲率的点，本质上是足够远离边界的简单连接的表面。这可以用来给出圆球的一个新的特征：圆球是嵌入欧氏空间中唯一具有非零常平均曲率的单连通曲面。这种特性是和谐的非常经典和优雅的结果，在理论中的常数平均曲率曲面所给予的霍普夫和亚历山德罗夫，以及最近和突破性的结果由Colding-Minicozzi和米克斯-罗森伯格。此外，知道这样的表面在欧几里德空间的几何构成坚实的基础，从那里开始调查的表面嵌入在一个3-流形与非零常数平均曲率。这也是项目的一部分。

This project consists of a thorough study of the geometry of nonzero constant mean curvature surfaces embedded in Euclidean space. One of the main goals of this project is obtaining some new and remarkable curvature estimates for simply-connected surfaces embedded in Euclidean space with nonzero constant mean curvature at points that are intrinsically sufficiently far away from the boundary. This can then be used to give a new characterisation of round spheres: round spheres are the only simply-connected surfaces embedded in Euclidean space with nonzero constant mean curvature. This characterisation is consonant to very classical and elegant results in the theory of constant mean curvature surfaces given by Hopf and Alexandrov, and more recent and groundbreaking results by Colding-Minicozzi and Meeks-Rosenberg. In addition, knowing the geometry of such surfaces in Euclidean space constitutes solid foundations from where to start an investigation of surfaces embedded in a 3-manifold with nonzero constant mean curvature. This is also part of the project.