Minimal Surfaces and Mean Curvature Flow

最小曲面和平均曲率流

• 批准号: 1404282
• 负责人: Brian White
• 金额: $ 23.5万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404282&HistoricalAwards=false
• 关键词: Minimal; Surfaces; Mean; Curvature; Flow

AbstractAward: DMS 1404282, Principal Investigator: Brian WhiteA minimal surface is the mathematical counterpart to a soap film spanning a curved wire. The elegant appearance of a film of soap is associated to the soap film's property of spanning the wire in the most efficient way possible, where efficiency here means using as little surface area as possible. The physical elegance of the soap film is reflected in the mathematical description of its efficiency, which was discovered in the 18th century and continues to motivate progress in geometry and differential equations. Among the topics to be pursued under the support of this grant are the properties of helicoid-like minimal surfaces, which spiral in space like the surface of a screw or a multi-story parking ramp; the classical helicoid was discovered in the 1770s to be a minimal surfaces, but a number of other examples that closely resemble helicoids in the large but are more complicated near the origin were discovered only in the last ten years.The principal investigator plans to study helicoid-like minimal surfaces, densities of minimal cones, the dependence of a minimal surface on the total curvature of its boundary, the branching behavior of minimal surfaces, and properties of mean curvature flow, particularly singularity formation and the non-uniqueness known as "fattening." The methods to be employed are a combination of classical minimal surface theory, geometric measure theory, and partial differential equations.

摘要奖：DMS 1404282，首席研究员：布莱恩·怀特极小曲面是横跨一条曲线的肥皂片的数学对应物。肥皂膜的优雅外观与肥皂膜以最有效的方式跨越导线的特性有关，在这里，效率意味着使用尽可能少的表面积。肥皂膜的物理优雅体现在对其效率的数学描述上，这一点在18世纪被发现，并继续推动几何和微分方程的进步。在这笔赠款的支持下，将研究的课题包括螺旋形极小曲面的性质，它在空间中像螺杆表面或多层停车坡道一样螺旋形；经典的螺旋面在17世纪70年代被发现是一个极小曲面，但在最近十年才发现了许多在大范围内与螺旋面非常相似但在原点附近更复杂的例子。主要研究者计划研究类似螺旋面的极小曲面、极小圆锥的密度、极小曲面对其边界总曲率的依赖、极小曲面的分支行为以及平均曲率流的性质，特别是奇点形成和被称为“肥化”的非唯一性。所采用的方法是经典极小曲面理论、几何测度论和偏微分方程组的组合。