Minimal surfaces and geometric flows

最小表面和几何流

• 批准号: 0906233
• 负责人: William Minicozzi
• 金额: $ 38.45万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906233&HistoricalAwards=false
• 关键词: Minimal; surfaces; geometric; flows

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.

PI建议，与Toby Colding一起，继续我们对极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。我们的分层定理和单边曲率估计在极小曲面上的一些结果中发挥了关键作用，包括螺旋面唯一性的Meek-Rosenberg证明，Calabi-Yau猜想的证明，以及Riemann例子唯一性的Meek-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了，正则性理论也是完备的，但本质上没有这样的界是已知的。根据曲面的拓扑结构，将研究分为三种情况。在曲面是圆盘的第一种情况下，我们的理解是相对完整的。主要问题是在更高亏格的情况下更好地理解紧性，理解势极限层的奇性(特别是类属性质)，以及理解像M.Weber，D.Hoffman和M.Wolf的亏格一螺旋面的空间这样的模空间(参见。与霍夫曼和B.怀特)。该提案的第二个广泛领域集中于对MCF中奇点的研究，包括对奇点集大小的估计、关于可能的奇点类型的紧致性定理以及流的一般奇点的分类。极小曲面是指局部最小化其表面积的曲面，因此是表面张力处于完美平衡且膜在其他时间不变的模型肥皂膜。至少自拉格朗日1762年的回忆录以来，极小曲面已经被研究过，但近年来在极小曲面理论中的许多长期存在的问题上取得了突破，许多数学家做出了重要贡献。有一个与此类似的时变模拟，其中一个曲面(不处于平衡状态)进化以尽可能快地最小化其表面积；这称为平均曲率流或MCF。在数学上，这导致了一个形式上类似于物理中控制热流的方程的非线性偏微分方程式。显然，最小曲面在MCF下保持静态。MCF和其他几何流是因为它们的内在美以及它们在其他领域的潜在应用而被开发出来的，例如，对期权定价、退火金属中的颗粒运动和晶体生长进行建模。虽然已经取得了关键的基础性结果，但一些最基本的问题仍然没有答案。