Mathematical Sciences: Mean Curvature Problems in Euclidean Space and Riemannian Analogs

数学科学：欧几里得空间和黎曼类比中的平均曲率问题

• 批准号: 8808002
• 负责人: Nicholas Korevaar
• 金额: $ 1.45万
• 依托单位: University of Kentucky
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1989-01-01
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8808002&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Mean; Curvature; Problems

Nicholas Korevaar will continue his work on surfaces of constant mean curvature in three dimensional space. Historically most research was directed at surfaces of zero mean curvature, these are known as minimal surfaces. Prototypical examples being soap films. During the last five years there has been a great deal of activity in the realm of non-zero mean curvature; here soap bubbles are the typical examples. This work received its impetus from Wente's counterexample to the Hopf conjecture. During the last year Kapouleas has shown how to construct many more such examples of different topological types. Korevaar's work is in many ways complementary to this and a natural development in the sense that he is looking at the structure that such surfaces must have rather than attempting to construct more examples. Korevaar's earlier work with his collaborators has led to a good understanding of the behaviour of these surfaces near to infinity. His attention will now turn to an investigation of the structure in bounded regions of space. This will involve an interaction of ideas from geometry and the theory of partial differential equations. He will also carry out related research into capillarity questions and into convexity theorems for solutions of elliptic equations.

Nicholas Korevaar将继续他的工作， 三维空间中的常数平均曲率。历史上 大多数研究针对零平均曲率的表面， 这些被称为最小表面。典型的例子是 肥皂电影在过去的五年里， 在非零平均曲率领域的活动;这里 肥皂泡是典型的例子。这项工作得到了其 从温特反例到霍普夫猜想的推动力。 在过去的一年里，Kapouleas展示了如何构建许多 不同拓扑类型的更多这样的示例。Korevaar的 工作在许多方面与此相辅相成， 发展的意义上说，他正在寻找的结构， 这样的表面必须有，而不是试图建立更多的 例子. Korevaar与他的合作者的早期工作已经导致了一个 很好地理解这些表面的行为， 无穷大.他的注意力现在将转向对 在有界空间区域中的结构。这将涉及一个 几何思想与局部理论的相互作用 微分方程他还将开展相关研究 毛细管现象和凸性定理， 椭圆方程的解