Mean Curvature Flow and Minimal Varieties

首席调查员将研究平均曲率流，这是一个几何形状随时间变化的自然过程。在数学中，平均曲率流和相似流已被证明是非常重要的：尤其是密切相关的Ricci流，因为它在解决长期存在的Poincare猜想中起着至关重要的作用，所以一直在新闻中非常重要。平均曲率流也出现在物理世界中。例如，退火金属中的晶界以平均曲率流动的方式移动。首席调查员还将调查最小曲面。最小曲面是平均曲率流的平衡形状，即不随时间变化的形状。在天文学中，最小的表面是黑洞的“视界”。在地球上，肥皂膜提供了极小表面的例子。极小曲面具有极大的理论意义：在极小曲面研究中首次发现的工具已被证明在数学的许多其他领域中都很有价值。首席研究员计划研究平均曲率流的性质，特别是奇点的形成和被称为“肥化”的非唯一性。具体目标包括确定普通曲面是否会产生比任意初始曲面更小的时空奇异集，了解肥胖的原因，以及发现防止肥胖的自然条件。他还计划研究极小簇，包括最近发现的亏格-g螺旋面在亏格趋于无穷时的行为，极小曲面的性质与其边界总曲率之间的关系，极小曲面分支点的精细结构(特别是极小曲面在分支点附近面积最小的Micallef-White必要条件是否也是充分的)，以及极小簇奇点上的拓扑和密度之间的关系。

The Principal Investigator will study mean curvature flow, a natural process by which geometric shapes change over time. In mathematics, mean curvature flow and similar flows have proved to be very important: in particular, the closely related Ricci flow has been very much in the news because of its essential role in the solution of the long-standing Poincare conjecture. Mean curvature flow also arises in the physical world. For example, grain boundaries in annealing metals move by mean curvature flow. The Principal Investigator will also investigate minimal surfaces. Minimal surfaces are equilibrium shapes for mean curvature flow, that is, shapes that do not change over time. In astronomy, minimal surfaces occur as "apparent horizons" of black holes. Here on earth, soap films provide examples of minimal surfaces. Minimal surfaces are of great theoretical interest: tools first discovered in the study of minimal surfaces have proved to be valuable in many other areas of mathematics.The Principal Investigator plans to study properties of mean curvature flow,especially singularity formation and the non-uniqueness known as "fattening". Specific goals include determining whether generic surfaces give rise to smaller spacetime singular sets than do arbitrary initial surfaces, understanding the causes of fattening, and discovering natural conditions that prevent fattening. He also plans to investigate minimal varieties, including topics such as the behavior of the recently-discovered genus-g helicoids as the genus tends to infinity, the relationship between properties of a minimal surface and the total curvature of its boundary, the fine structure of branch points of minimal surfaces (and, in particular, whether the Micallef-White necessary conditions for a minimal surface to be area-minimizing near a branch point are also sufficient), and the relationship between topology and density at singularities of minimal varieties.