Non-Compact Solutions to Geometric Flows

几何流是通过曲面或高维空间的曲率演化的过程。具体地说，本项目中考虑的流动是微分方程式，它模拟了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速地减少而变化。例如，在平均曲率流(MCF)下，表面积减小最快，而在高斯曲率流(GCF)下，封闭体积减小最快。由于这些自然的递减特性，随着流动变得奇异(即，对象发展成褶皱、拐角或其他高曲率的点)，这些演化通常趋向(在放大下)使诸如面积和体积之类的相应能量最小化的最佳形状。例如，肥皂泡是MCF的重新定标奇点的形状。这些能量最小化不仅出现在几何学中，而且还出现在经济学和物理学中；例如，最佳运输指的是从一个地区到另一个地区的映射，该映射将能源最小化，这是资源分配的总成本。因此，研究几何流的奇异性对于理解物理和经济中的能量极小化有了新的认识。本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Ricci流(RF)。对于MCF和RF，将考虑非折叠第二类古老解的唯一性。对于GCF，将在适当的条件下研究第二类封闭古解的存在唯一性。此外，本项目还考察了曲线缩短流和GCF的平移解的收敛问题。将开发内部估计、衰减率和单调性公式。此外，该项目还将解决GCF和其他完全非线性方程的最优正则性和自由边界问题，包括最优传输和Monge-Ampere方程。这个项目将提供一种方法，利用规定的奇点条件来获得最优的正则性和自由边界正则性，特别是在从数量经济学和经典力学产生的自由边界问题中。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.