Minimal surfaces and geometric analysis

最小曲面和几何分析

• 批准号: 0405695
• 负责人: William Minicozzi
• 金额: $ 43.2万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405695&HistoricalAwards=false
• 关键词: Minimal; surfaces; geometric; analysis

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.

MS-0405695题目：极小曲面和几何分析研究者：William P. Minicozzi，约翰霍普金斯大学摘要我们将继续我们对极小曲面和几何分析的相关领域的研究，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--叠层定理和单侧曲率估计--都是关于适当嵌入的极小圆盘的。 近年来，在许多数学家的贡献下，极小曲面理论的许多长期问题都取得了突破性进展。 这里描述的叠层定理和单侧曲率估计起了关键作用，并被许多人使用。 其中两个重要的新方向是去除适当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影响。 最小曲面领域的历史可以追溯到早期工作的欧拉在1744年和拉格朗日在1762年，并一直保持一个充满活力的研究领域，为过去的250年。 最小表面在科学中经常出现，至少可以追溯到世纪上半叶比利时物理学家普拉托的肥皂膜实验。 他们的数学影响是显着的，并导致了发展几何，拓扑学和偏微分方程。 这个主题最近有了重大的发展，包括对一些长期悬而未决的问题的回答，以及对适当嵌入的最小圆盘（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，答案可能会导致进一步的发展。