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性、如何退化以及它们的莫尔斯指数是否有界。这些结果有沿着许多可能的应用，包括球面空间形式问题，具有正数量曲率的三维流形的拓扑等，此外，这些结果将对欧氏空间中的极小曲面有新的认识。 最小表面是面积的关键点（例如，肥皂膜是最小面积的表面），在数学和其他科学的许多问题中自然出现。 这方面的许多经典结果都是在最小面积曲面上得到的。 相比之下，我们的主要兴趣是高指数（即，高度不稳定）嵌入极小曲面与有界拓扑-一个著名的例子是由标准螺旋面。 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.