Non-Compact Solutions to Geometric Flows

几何流是曲面或高维空间通过曲率演化的过程。特别地，在这个项目中考虑的流动是微分方程，它模拟了几何物体的形状如何随着其面积、体积或其他一些几何量尽可能快地减少而变化。例如，在平均曲率流（MCF）下，表面积减小得最快，而在高斯曲率流（GCF）下，封闭体积减小得最快。由于这些自然的递减特性，当流动变得奇异（即，物体产生褶皱、角落或其他高曲率点）时，这些演变通常（在放大下）趋向于使相应的能量（如面积和体积）最小化的最佳形状。例如，肥皂泡是MCF的一个重新缩放的奇点的形状。这些能量最小化不仅出现在几何中，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，最小化能源，这是资源分配的总成本。因此，对几何流奇异性的研究，为研究几何流的奇异性提供了新的思路。了解能源最小化吗？在物理和经济学方面。本项目旨在了解高斯曲率流（GCF）、平均曲率流（MCF）和里奇流（RF）等几何流的奇异性。对于MCF和RF，将考虑非坍塌型II型古解的唯一性。对于GCF，将在合适的条件下研究II型闭古解的存在性和唯一性。此外，本项目还研究了曲线缩短流和GCF的平移解的收敛性。内部估计，衰减率和单调性公式将开发。此外，该项目还将解决GCF和其他全非线性方程的最优正则性和自由边界问题，包括最优输运和蒙日-安培方程。本项目将提供一种利用规定的奇点条件来获得最优正则性和自由边界正则性的方法，特别是在定量经济学和经典力学中出现的自由边界问题中。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.