Minimal Surfaces and Mean Curvature Flow

获奖：DMS 1404282，首席研究员：Brian white最小表面是横跨弯曲导线的肥皂膜的数学对应。肥皂膜的优雅外观与肥皂膜以最有效的方式跨越电线的特性有关，这里的效率意味着使用尽可能小的表面积。肥皂膜的物理优雅体现在对其效率的数学描述中，这是在18世纪发现的，并继续推动几何和微分方程的进步。在这项拨款的支持下，将进行的主题包括螺旋状最小表面的性质，这些表面在空间中像螺旋或多层停车坡道的表面一样旋转；经典的螺旋面是在18世纪70年代被发现的，是一个最小的曲面，但是许多其他的例子，在大范围上与螺旋面非常相似，但在原点附近更复杂，直到最近十年才被发现。首席研究员计划研究类螺旋曲面最小曲面，最小锥密度，最小曲面对其边界总曲率的依赖，最小曲面的分支行为，以及平均曲率流的性质，特别是奇点形成和非唯一性被称为“增肥”。所采用的方法是经典最小曲面理论、几何测度理论和偏微分方程的结合。

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.