Elliptic and Parabolic Partial Differential Equations on Manifolds

我们用来理解物理理论的主要数学对象是偏微分方程式。有许多这样的方程，它们的解的行为反映了我们观察到的不同类型的现象，包括热的分散、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组来理解几何空间和可能存在于几何空间上的结构。本课题研究了偏微分方程组在两类几何问题中的应用。第一个涉及Calabi-Yau方程：这是一种在弦论中用作模型的偏微分方程组，在研究由代数方程定义的几何空间中有广泛的应用。这个项目的一个目标是推广和求解结构少得多的空间上的Calabi-Yau方程，长期目标是对所有这样的空间进行分类。第二类几何问题涉及一种称为坍塌的现象。这发生在几何热方程的研究中，在该方程中，几何空间在时间上演化，并可能在某些方向上坍塌，从而产生较低维度的对象。这种塌陷可以揭示原始空间的结构。为了开展这些调查，国际和平研究所需要开发新的技术工具。PI将利用已为经典方程(如热方程)开发的技术，并将使其适用于几何中出现的非线性偏微分方程组的研究。本项目将研究非线性椭圆型和抛物型方程，并应用于复杂和几乎几何。特别是，PI将研究在流形上规定体积形式的问题，推广了著名的Yau关于紧致Kahler流形的定理。基于PI在Hermitian流形和Gauduchon流形上的工作，PI将研究为平衡度量和四维流形上的几乎Kahler度量规定体积形式的问题。这个项目的另一个主要目标是了解沿几何流动崩塌的现象。Kahler-Ricci流在无限时间的坍塌现在已经被很好地理解了。PI将专注于有限时间崩溃的难题。这发生在Fano流形上的Kahler-Ricci流和非Kahler复流形上的Chern-Ricci流(厄米度规的流)，例如Hopf曲面上。为了实现这些目标，PI将为研究非线性偏微分方程组开发新的工具。特别是，PI将考虑利用黑森最大特征值的凸性的新的二阶估计。这些估计已经成功地用于建立一类一般偏微分方程常数秩定理和退化复Monge-Ampere方程的最优正则性结果。PI还将发展多点最大值原理，该原理在复杂几何背景下研究偏微分方程组的解的凸性性质方面有很长的历史。

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.