Fully nonlinear elliptic equations in geometry

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Differential equations play a central role in mathematics and its applications, for example, in differential geometry and mathematical physics. Fully nonlinear elliptic and parabolic equations are some of the main equations under current investigations by researchers. Building on the recent progress, in this project the principal investigator will establish general existence and regularity results for these equations on real or complex manifolds and develop technical tools which can be applied to solve other types of equations. Results in this project will yield interesting applications in geometry and help to understand general partial differential equations.A central issue in solving fully nonlinear elliptic and parabolic equations is to establish a priori estimates which present great challenges due to the high nonlinearity of the equations and the general geometry of the manifolds as well as their boundaries. The project's goal is to search for new methods to overcome these difficulties. In particular, it will identify the optimal conditions under which one can solve the equations on general manifolds. The principal investigator also plans to study a very general class of equations on which few results are known, and attack some difficult problems in complex geometry.