Non-linear partial differential equations in geometry

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In this project, the principal investigators will study several higher-order nonlinear partial differential equations that arise in conformal geometry and Cauchy-Riemann (CR) geometry. These equations describe the effect on the curvatures of the new structures of making conformal changes of metrics (as in conformal geometry) or of contact forms (as in CR-geometry). Of particular interest are equations that prescribe higher-order curvature invariants that control the global geometry of higher dimensional manifolds. The project seeks to develop new tools to solve these equations and to describe the behavior of the solutions. A key difficulty in handling such equations is the absence of a maximum principle, and the principal investigators propose to compensate for its absence by using integral inequalities as the main working tool. It is expected that the study will yield extensions of the well-known mixed volume inequalities to more general domains in Euclidean space, as well as sharp Sobolev inequalities in CR-geometry.The goal of this project is to provide new insights into and to create new tools for the study of the theory of geometric partial differential equations, a subject that has evolved into a basic toolbox in many areas in mathematics, applied mathematics, and engineering, to say nothing of mathematical physics. One of the main topics of study in the project (namely, the existence of so-called Einstein space) is motivated by a conjecture that physicists call the "holography principle." It asserts that the results of any measurements of the physical universe in the Einstein space can be predicted by taking another set of measurements near the "far end" of the universe (which is referred to as its "boundary"). A case of particular interest is the situation when the boundary space is three-dimensional, a case in which geometric understanding is already well developed. The project will engage graduate students and postdocs in its research activities.