Minimal surfaces and geometric analysis

标题：极小曲面和几何分析PI：约翰·霍普金斯大学的William P.Minicozzi摘要我们将继续研究极小曲面和几何分析的相关领域，包括几何演化方程，如平均曲率和Ricci流，以及函数论。到目前为止，我们关于极小曲面的一些主要结果--层合定理和单边曲率估计--是关于适当嵌入的极小圆盘的。近年来，极小曲面理论在许多长期存在的问题上取得了突破，许多数学家做出了重要贡献。本文所述的层合定理和单边曲率估计起到了关键作用，并已被许多人使用。其中两个重要的新方向是去掉充分性假设和考虑具有更一般拓扑类型的极小曲面。这些结果将产生重要的影响。极小曲面的研究可以追溯到欧拉在1744年和拉格朗日在1762年的早期工作，在过去的250年里一直是一个充满活力的研究领域。微小表面在整个科学中经常出现，至少可以追溯到19世纪上半叶比利时物理学家高原的肥皂膜实验。它们在数学上的影响很大，并导致了几何学、拓扑学和偏微分方程式的发展。该课题最近取得了重大进展，包括对一些长期悬而未决的问题的解答，以及对适当嵌入的最小圆盘(例如最初于1776年发现的螺旋面)的相当完整的描述。然而，当这些假设被移除时，我们所知的要少得多；理解这一点是我们研究的关键部分，而答案可能会导致进一步的发展。

MS-0405695Title: Minimal surfaces and geometric analysisPI: William P. Minicozzi, Johns Hopkins UniversityABSTRACTWe will continue our investigations on minimal surfaces and related areas of geometric analysis, including geometric evolution equations such as the meancurvature and Ricci flow, and on function theory. Some of ourmain results for minimal surfaces so far - the lamination theoremand the one-sided curvature estimate - are for properly embeddedminimal disks. Recent years have seen breakthroughs on many long--standingproblems in the theory of minimal surfaces, with importantcontributions from many mathematicians. The lamination theoremand the one-sided curvature estimate described here have played akey role and have been used by many people. Two of the important new directions are removing the assumption of properness and considering minimal surfaces with more general topological types. These results will have important implications. The field of minimal surfaces dates back to early work of Euler in1744 and Lagrange in 1762 and has remained a vibrant area of researchfor the last 250 years. Minimal surfaces appear frequently throughout science, dating back at least to the soap film experiments of the Belgian physicist Plateau in the first half of the nineteenth century. Their mathematical impact has been significant and has led to developments in geometry, topology, and partial differential equations. The subject has seen major developments recently, including answers to some long-standing open questions and a rather complete picture for properly embedded minimal disks (such as the helicoid which was originally discovered in1776). However, much less is known when these assumptions are removed; understanding this is a key part of our research and the answers are likely to lead to further developments.