Non-Compact Solutions to Geometric Flows

几何流的非紧解

• 批准号: 1811267
• 负责人: Tobias Colding
• 金额: $ 16万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811267&HistoricalAwards=false
• 关键词: Non; Compact; Solutions; Geometric; Flows

Geometric flows are processes that evolve surfaces or higher-dimensional spaces by their curvatures. In particular, the flows considered in this project are differential equations that model how the shape of a geometric object changes as its area, volume, or some other geometric quantity decreases as rapidly as possible. For example, the surface area decreases most rapidly under the mean curvature flow (MCF), while the enclosed volume deceases most rapidly under the Gauss curvature flow (GCF). Due to these natural decreasing properties, as the flow becomes singular (i.e., the object develops folds, corners, or other points of high curvature) these evolutions often tend (under magnification) toward optimal shapes minimizing the corresponding energies such as area and volume. For example, a soap bubble is the shape of a rescaled singularity of the MCF. These energy minimizers appear not only in geometry, but also in economics and physics; for instance, optimal transport refers to a mapping from one area to another that minimizes an energy which is the total cost of resource allocation. Thus, studying singularity of the geometric flows sheds new insight on the?understanding of energy minimizers?in physics and economics.This project aims to understand the singularity of various geometric flows including the Gauss curvature flow (GCF), the mean curvature flow (MCF), and the Ricci flow (RF). For the MCF and the RF, the uniqueness of non-collapsed type II ancient solutions will be considered. For the GCF, the existence and the uniqueness of type II closed ancient solutions will be studied under suitable conditions.?Moreover, this project also examines the convergence to the translating solutions to the curve-shortening flow and the GCF.? Interior estimates, decay rate, and monotonicity formulas will be developed.?In addition, this project will also address optimal regularity and free boundary problems for the GCF and and other fully non-linear equations, including optimal transport and Monge-Ampere equations. This project will provide a method to utilize prescribed singularity conditions to obtain optimal regularity and free boundary regularity, in particular in free boundary problems that arise from quantitative economics and classical mechanics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

几何流是通过曲面或高维空间的曲率来演化曲面或高维空间的过程。 特别地，在这个项目中考虑的流动是微分方程，其模拟几何对象的形状如何随着其面积、体积或其他几何量尽可能快地减小而变化。例如，表面积在平均曲率流（MCF）下减小最快，而封闭体积在高斯曲率流（GCF）下减小最快。 由于这些自然减少的特性，当流动变得奇异时（即，物体产生折叠、拐角或其它高曲率点），这些演变通常（在放大下）趋向于最佳形状，从而最小化相应的能量，例如面积和体积。 例如，肥皂泡是MCF的重新缩放奇点的形状。这些能量最小化者不仅出现在几何学中，也出现在经济学和物理学中;例如，最优运输是指从一个区域到另一个区域的映射，最小化能量，这是资源分配的总成本。因此，研究几何流的奇异性，为研究几何流的性质提供了新的思路。对能量最小化的理解物理学和经济学。该项目旨在了解各种几何流的奇异性，包括高斯曲率流（GCF），平均曲率流（MCF）和里奇流（RF）。对于MCF和RF，将考虑非塌陷的II型古解的唯一性。在适当的条件下，研究了GCF的第II型闭古解的存在唯一性.此外，本项目还研究了曲线缩短流和GCF的翻译解决方案的收敛性。内部估计，衰减率和单调性公式将被开发。此外，该项目还将解决全球气候变化框架的最佳规则性和自由边界问题，以及其他完全非线性方程，包括最佳运输和蒙格-安培方程。该项目将提供一种方法，利用规定的奇异性条件，以获得最佳的正则性和自由边界正则性，特别是在自由边界问题，产生于定量经济学和经典力学。该奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。