Geometric nonlinear partial differential equations

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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. A common thread linking our program is the analysis of the geometric fully nonlinear equations and their relationship with geometric quantities . The fundamental existence and regularity questions in the category of analysis should be adapted to cope with the emphasis on {\it geometric} solutions, which are often forced upon us by the geometric nature of the problems like the monotonicity of specified geometric functionals. Our objective is to develop various analytic tools to establish a priori estimates for these equations, explore the structures of geometric solutions, and derive geometric consequences.