Geometric nonlinear partial differential equations

微分几何和微分方程组涉及到许多实际问题，在物理和工程等领域中往往具有很强的理论性。这些问题大多是由非线性微分方程组来指导的。例如，理论物理中的弦理论，其中微分几何和微分方程扮演着重要的角色。在寻找具有某些几何性质的微分方程解的过程中使用了各种分析思想，特别是那些非线性的微分方程解。微分几何的主要内容是寻找“最优”几何结构，如微分同胚、度量等，并研究它们存在的几何和拓扑含义。这些“极值”几何对象可以看作是自然非线性偏微分方程解，它们编码了联系几何、拓扑和分析的丰富信息。曲率张量产生了重要的例子，例如Ricci张量和Weingarten映射。这些曲率张量的研究一般是通过抛物型和椭圆型非线性方程组来进行的。这些例子包括Monge-Amp方程、Gauss曲率流和Ricci流。我们继续研究与微分几何问题相关的完全非线性偏微分方程解。

Differential geometry and differential equations are involved in many practical problems and often highly theoretical problems in Physics and Engineering and other areas. Most of these problems are guided by nonlinear differential equations. For example, the string theory in theoretical physics, where differential geometry and differential equations play important roles. Various ideas of analysis are used in the process of finding solutions of differential equations with certain geometric properties in general and in particular those that are nonlinear. The main thrusts in differential geometry are the search for ``optimal" geometric structures, such as diffeomorphisms, metrics, etc., and the study of the geometric and topological implications of their existence. These ``extremal" geometric objects can be viewed as solutions to natural nonlinear partial differential equations, and they encode rich information linking geometry, topology and analysis. Curvature tensors yield important examples, e.g. Ricci tensor and Weingarten map. The study of these curvature tensors in general carried out through systems of parabolic and elliptic nonlinear equations. The examples including the Monge-Amp\`ere equations, Gauss curvature flow, and the Ricci flow. We continue to study the fully nonlinear partial differential equations related to problems in differential geometry.