Embedded Minimal Surfaces in Three Manifolds

我们提出了研究三维流形中嵌入极小曲面的各种方向。这包括研究这类曲面的收敛、它们如何退化以及它们的Morse指数是否有界。这些结果有许多可能的应用，包括球面空间型问题，具有正标量曲率的三维流形的拓扑等。此外，这些结果应该是对欧几里德空间中极小曲面的新的见解。本项目主要关注三维流形中的极小曲面。极小曲面是面积的临界点(例如，肥皂膜是最小面积曲面)，在数学和其他科学中的许多问题中自然产生。这一领域的许多经典结果都是在最小面积曲面上得到的。相反，我们主要感兴趣的是具有有界拓扑的高指数(即高度不稳定)嵌入极小曲面--标准螺旋面给出了一个著名的例子。粗略地说，我们的一些定理的要点是螺旋面的行为是典型的。

We propose various directions for studying embedded minimal surfacesin three-manifolds. This includes the study of the convergence of suchsurfaces, how they degenerate and whether or not their Morse index isbounded. There are many possible applications of results along these linesincluding to the spherical space-form problem, the topology ofthree-manifolds with positive scalar curvature etc. In addition, these results shouldyield new insight into minimal surfaces in Euclidean space.This project focuses on minimal surfaces in three-dimensionalmanifolds. Minimal surfaces are critical points for area (e.g.,soap films are least area surfaces) and arise naturally in manyproblems in mathematics and in the other sciences. Many of theclassical results in this area are on least area surfaces. In contrast,our main interest is in high index (i.e., highly unstable) embedded minimalsurfaces with bounded topology - a well-known example is given bythe standard helicoid. Roughly speaking, the point of some of our theoremsis that the behavior of the helicoid is typical.