Hermitian流形上的完全非线性偏微分方程

Many problems in geometry can be worked out by solving related partial differential equations on Hermitian manifolds. In this project, we study Hessian type partial differential equations of general form on Hermitian manifolds. These equations, particularly a special case named Monge-Ampère equations, are a class of fully nonlinear partial differential equations which play an important role in Geometric Analysis. They appear in many problems in geometry and theoretical physics including Calabi conjecture, Chern-Levine-Nirenberg conjecture, Gauduchon conjecture, solving the Strominger system and so on. In the project we shall consider Hessian type equations both on Hermitian manifolds with bounary and without boundary (which are called closed Hermitian manifolds) and we are mainly concerned with the a priori estimates for these equations for which we shall study the geometrical properties of level sets of symetric concave functions. We are also interested in that what geometric condition can ensure the existence of solutions. In particular, we shall consider a conjecture about this question proposed by Székelyhidi in 2015.

几何中很多问题可以通过求解Hermitian流形上的偏微分方程来解决。本项目将研究Hermitian流形上具有一般形式的Hessian型方程。这类方程（特别是作为它特殊情形的Monge-Ampère型方程）是几何分析中一类重要的完全非线性偏微分方程 ，它出现在许多几何问题和理论物理问题中，如Calabi猜想、Chern-Levine-Nirenberg猜想、Gauduchon猜想、Strominger系统的求解等。在本项目中，我们分别考虑带边Hermitian流形和不带边Hermitian流形（称之为闭Hermitian流形）上的Hessian型方程两种情形，并且重点研究这类方程的先验估计，为此，我们将深入研究对称凹函数的水平集的几何性质。本项目还将探索能够保证解存在的几何条件，特别地，我们将研究Székelyhidi在2015年提出的关于此问题的一个猜测。

流形上的完全非线性方程出现在许多重要的几何问题和理论物理问题中，如Calabi猜想、Chern-Levine-Nirenberg猜想、Gauduchon猜想、Strominger系统的求解等，这些方程通常是Monge-Ampère型的。我们的目的是构建完全非线性方程的一般性理论。我们首先研究了一类Hessian型方程的先验估计，并得到了容许经典解的存在性。接下来，我们考虑了关于退化k-Hessian方程的Dirichlet问题广义解正则性的一个公开问题，在附加了一个条件后得到了解的先验估计，从而取得了一定进展。最后，我们研究了流形上的一类完全非线性偏微分方程，得到了容许解的二阶先验估计。