Minimal surfaces and geometric flows

PI提议与Toby Colding一起，继续我们对最小表面和几何分析相关领域的研究，包括几何演化方程，如平均曲率（MCF）和里奇流。我们的层合定理和单侧曲率估计在最小曲面上的许多结果中发挥了关键作用，包括meeks - rosenberg对螺旋面唯一性的证明，我们对Calabi-Yau猜想的证明，以及Meeks-Perez-Ros对黎曼例子唯一性的证明。具有均匀曲率（或面积）边界的最小曲面已经被很好地理解，并且正则性理论是完整的，但是基本上没有这样的边界就什么都不知道。该研究根据表面的拓扑结构分为三种情况。在第一种情况下，当表面是圆盘时，我们的理解是相对完整的。主要问题是更好地理解高属情况下的紧性，理解潜在极限层合的奇点（特别是一般性质），以及理解M. Weber、D. Hoffman和M. Wolf（参见Hoffman和B. White）的一类螺旋形的模空间。该提案的第二个广泛领域集中在对MCF中的奇点的研究上，包括对奇异集大小的估计，奇点可能类型的紧性定理，以及流的一般奇点的分类。最小表面是局部表面积最小的表面，因此，模型肥皂膜的表面张力处于完美的平衡状态，并且膜在其他时间不会改变。它们至少从拉格朗日1762年的回忆录开始就被研究了，但近年来，在许多数学家的重要贡献下，极小曲面理论中许多长期存在的问题取得了突破。这有一个时变的模拟，即一个表面（不处于平衡状态）演变成尽可能快地最小化其表面积；这被称为平均曲率流或MCF。从数学上讲，这导致了一个非线性偏微分方程，它在形式上类似于物理学中控制热量流动的方程。显然，最小的表面在MCF下保持静态。MCF和其他几何流因其内在之美以及它们在其他领域的潜在应用而被开发出来。用于建模，例如，期权定价，退火金属中晶粒的运动和晶体生长。虽然已经获得了关键的基础结果，但几个最基本的问题仍未得到解答。

The PI proposes, jointly with Toby Colding, to continue ourinvestigations on minimal surfaces and related areas of geometricanalysis, including geometric evolution equations such as the meancurvature (MCF) and Ricci flow. Our lamination theorem and one-sided curvature estimateplayed a key role in a number of results on minimal surfaces including theMeeks-Rosenberg proof of uniqueness of the helicoid, our proofof the Calabi-Yau Conjectures, and the Meeks-Perez-Ros proof ofthe uniqueness of the Riemann examples. Minimal surfaces with uniform curvature (or area) bounds have beenwell understood and the regularity theory is complete, yetessentially nothing was known without such bounds. The study isdivided into three cases depending on the topology of the surface. Our understanding is relatively complete in the first case when the surface is a disk.The main problems are to get a finer understanding of compactness in the higher genus cases, to understand singularities of potential limit laminations (especially generic properties), and to understand moduli spaces like the space of genus one helicoids of M. Weber, D. Hoffman and M. Wolf (cf. with Hoffman and B. White). The second broad area of the proposal centers on the study of singularities in MCF, including estimates for the size of the singular sets, a compactness theorem for the possible types of singularities, and a classification of generic singularities of the flow. Minimal surfaces are surfaces that locally minimize their surface area and, thus, model soap films where surface tension is in perfect equilibrium and the film does not change other time.They have been studied at least since Lagrange's 1762 memoir, but recent years have seenbreakthroughs on many long--standing problems in the theory ofminimal surfaces, with important contributions from manymathematicians. There is a time-varying analog of this where a surface (which is not in equilibrium) evolves to minimize its surface area as quickly as possible; this is called mean curvature flow or MCF. Mathematically, this leads to a nonlinear partial differential equation which is formally similar to the equation that governs the flow of heat in physics. Clearly, minimal surfaces remain static under the MCF. MCF and other geometric flows were developed for their intrinsic beauty as well as their potential applications to other fields to model, for instance, option pricing, motion of grains in annealing metals, and crystal growth.While key foundational results have been obtained, several of the most basic questions remain unanswered.