Minimal surfaces and geometric flows

PI建议，与Toby Colding一起，继续我们对最小曲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Ricci流。 我们的叠层定理和单侧曲率估计在极小曲面的许多结果中起了关键作用，包括螺旋面唯一性的Meeks-Rosenberg证明， 我们对Calabi-Yau猜想的证明，以及Riemann例子的唯一性的Meeks-Perez-Ros证明。 具有一致曲率（或面积）界的极小曲面已经被很好地理解，正则性理论也已完备，但实际上没有这样的界是什么都不知道的。 根据曲面的拓扑结构，将研究分为三种情况. 我们对第一种情形（曲面为圆盘）的理解是比较完整的，主要问题是更好地理解高阶亏格情形下的紧性，理解位势极限叠层的奇异性（特别是一般性质），以及理解模空间（如M的亏格为1的螺旋面空间）。Weber，D.霍夫曼和M.狼（cf.和霍夫曼还有B在一起白色）。 该提案的第二个广泛领域集中在MCF中奇点的研究上，包括奇异集的大小估计，可能的奇点类型的紧性定理，以及流的一般奇点的分类。 极小曲面是表面，局部最小化其表面积，因此，模型肥皂膜的表面张力是在完美的平衡和电影不改变其他时间。他们一直在研究至少自拉格朗日的1762年回忆录，但近年来已经看到突破了许多长期存在的问题，在理论ofminimal曲面，与重要贡献，从许多数学家。 有一个随时间变化的模拟 其中一个表面（不处于平衡状态）尽可能快地发展以最小化其表面积;这被称为平均曲率流或MCF。 在数学上，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流的方程。 显然，最小表面在MCF下保持静止。MCF 和其他几何流 开发其内在的美丽，以及其潜在的应用， fi例如，期权定价、退火金属中晶粒的运动和晶体生长等。虽然已经获得了关键的基础性结果，但仍有几个最基本的问题没有得到解答。

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.