Elliptic and Parabolic Partial Differential Equations on Manifolds

流形上的椭圆和抛物型偏微分方程

基本信息

批准号：
1709544
负责人：
Benjamin Weinkove
金额：
$ 20.7万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709544&HistoricalAwards=false
关键词：
Elliptic Parabolic Partial Differential Equations

项目摘要

The principal mathematical objects we use to understand physical theories are partial differential equations (PDEs). There are many such equations, and the behavior of their solutions reflects the different kinds of phenomena we observe including the dispersion of heat, the effect of gravity, the motion of fluids and the movement of subatomic particles. Moreover, mathematicians use PDEs to understand geometric spaces and the possible structures that can exist on them. This project investigates the use of PDEs in two kinds of geometric problems. The first concerns the Calabi-Yau equation: this is a PDE used as a model in string theory and has wide-ranging applications in the study of geometric spaces defined by algebraic equations. A goal of this project is to generalize and solve the Calabi-Yau equation on spaces with much less structure, with a long term aim of classifying all such spaces. The second kind of geometric problem concerns a phenomenon known as collapsing. This occurs in the study of geometric heat equations where a geometric space evolves in time and may collapse in some directions to yield a lower dimensional object. This collapsing can reveal the structure of the original space. In order to carry out these investigations, the PI will need to develop new technical tools. The PI will take advantage of techniques which have been developed for classical equations such as the heat equation, and will adapt them to the study of non-linear PDEs occurring in geometry.This project will investigate nonlinear elliptic and parabolic equations, with applications to complex and almost geometry. In particular, the PI will study the problem of prescribing volume forms on manifolds, extending the well-known theorem of Yau for compact Kahler manifolds. Building on the PI's work on Hermitian and Gauduchon manifolds, the PI will investigate the question of prescribing volume forms for balanced metrics, and for almost Kahler metrics on four-manifolds. Another major goal of this project is to understand the phenomenon of collapsing along geometric flows. Collapsing for the Kahler-Ricci flow at infinite time is now quite well-understood. The PI will focus on the difficult problem of finite time collapse. This occurs for the Kahler-Ricci flow on Fano manifolds and also for the Chern-Ricci flow (a flow of Hermitian metrics) on non-Kahler complex manifolds such as the Hopf surface. To accomplish these goals, the PI will develop new tools for the study of nonlinear PDE. In particular, the PI will consider new second order estimates exploiting the convexity of the largest eigenvalue of Hessian. These kinds of estimates have already been used successfully to establish constant rank theorems for a general class of PDEs and optimal regularity results for the degenerate complex Monge-Ampere equation. The PI will also develop multi-point maximum principles, which have a long history in the study of convexity properties of solutions to PDEs, in the context of complex geometry.

我们用来理解物理理论的主要数学对象是部分微分方程（PDE）。有许多这样的方程式，其溶液的行为反映了我们观察到的不同种类的现象，包括热的分散，重力的影响，流体运动和亚原子颗粒的运动。此外，数学家使用PDE来了解几何空间以及可能存在的可能存在的结构。该项目研究了在两种几何问题中使用PDE的。第一个涉及Calabi-yau方程：这是一个PDE用作字符串理论中的模型，并且在代数方程定义的几何空间中具有广泛的应用。该项目的一个目标是在结构少得多的空间上概括和求解卡拉比YAU方程，长期目的是对所有此类空间进行分类。第二种几何问题涉及一种称为崩溃的现象。这发生在几何热方程的研究中，其中几何空间随着时间的流逝而演变，并且可能会在某些方向上塌陷以产生较低的尺寸对象。这种崩溃可以揭示原始空间的结构。为了进行这些调查，PI需要开发新的技术工具。 PI将利用针对经典方程（例如热方程）开发的技术，并将它们适应几何形状中发生的非线性PDE的研究。该项目将研究非线性椭圆形和抛物线方程，并应用于复杂和几乎几何形状。特别是，PI将研究在歧管上规定体积形式的问题，从而将Yau的著名定理扩展到紧凑的Kahler歧管。 PI以Pi在Hermitian和Gauduchon流形的作品为基础，PI将调查针对平衡指标的规定数量形式的问题，以及几乎在四个manifolds上的Kahler指标。该项目的另一个主要目标是了解沿几何流量崩溃的现象。现在，无限时间的Kahler-Ricci流崩溃了。 PI将重点放在有限的时间崩溃的困难问题上。这是针对Fano歧管上的Kahler-Ricci流以及非Kahler复合物歧管（例如Hopf表面）上的Chern-Ricci流（Hermitian指标流）的情况。为了实现这些目标，PI将开发用于研究非线性PDE的新工具。特别是，PI将考虑利用Hessian最大特征值的凸度的新的二阶估计。这些类型的估计值已经成功地用于建立一类PDE类别的恒定等级定理，并为退化的复杂复杂的Monge-Ampere方程提供了最佳的规律性结果。 PI还将开发多点最大原则，在复杂的几何形状的背景下，在溶液对PDES的凸特性的研究中具有较长的历史。