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偏微分方程（PDE）。这样的方程有很多，它们的解的行为反映了我们观察到的不同类型的现象，包括热量的扩散、重力的影响、流体的运动和亚原子粒子的运动。此外，数学家使用偏微分方程来理解几何空间及其上可能存在的结构。该项目研究偏微分方程在两种几何问题中的应用。第一个涉及 Calabi-Yau 方程：这是用作弦理论模型的偏微分方程，在代数方程定义的几何空间的研究中具有广泛的应用。该项目的目标是在结构较少的空间上推广和求解 Calabi-Yau 方程，长期目标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塌陷的现象。这种情况发生在几何热方程的研究中，其中几何空间随时间演化，并且可能在某些方向上塌陷以产生较低维度的物体。这种倒塌可以揭示原始空间的结构。为了进行这些调查，PI 将需要开发新的技术工具。 PI 将利用为经典方程（例如热方程）开发的技术，并将其应用于几何中非线性偏微分方程的研究。该项目将研究非线性椭圆方程和抛物线方程，并将其应用于复杂的方程和几乎几何。特别是，PI 将研究在流形上规定体积形式的问题，将著名的丘定理扩展到紧致卡勒流形。以 PI 在埃尔米特和高杜雄流形方面的工作为基础，PI 将研究为平衡度量以及四流形上的几乎卡勒度量规定体积形式的问题。该项目的另一个主要目标是了解沿几何流塌陷的现象。现在，卡勒-里奇流在无限时间内的塌陷已得到很好的理解。 PI 将重点关注有限时间崩溃这一难题。这种情况发生在 Fano 流形上的 Kahler-Ricci 流以及非 Kahler 复流形（例如 Hopf 表面）上的 Chern-Ricci 流（埃尔米特度量流）上。为了实现这些目标，PI 将开发新的非线性偏微分方程研究工具。特别是，PI 将考虑利用 Hessian 最大特征值的凸性进行新的二阶估计。此类估计已成功用于建立一般类偏微分方程的常秩定理以及简并复数 Monge-Ampere 方程的最佳正则性结果。 PI 还将开发多点极大值原理，该原理在复杂几何背景下偏微分方程解的凸性研究方面有着悠久的历史。