Non-linear partial differential equations in geometry

在这个项目中，主要研究人员将研究在共形几何和柯西-黎曼(CR)几何中出现的几个高阶非线性偏微分方程组。这些方程描述了度量的共形改变(如在保角几何中)或接触形式(如在CR几何中)对新结构曲率的影响。特别令人感兴趣的是规定了控制高维流形全局几何的高阶曲率不变量的方程。该项目寻求开发新的工具来求解这些方程并描述解的行为。处理这类方程的一个关键困难是没有最大值原理，主要调查员建议用积分不等式作为主要工作工具来弥补它的缺失。这个项目的目标是为几何偏微分方程理论的研究提供新的见解和新的工具。几何偏微分方程组已经发展成为数学、应用数学和工程学中许多领域的基本工具箱，更不用说数学物理了。该项目的主要研究课题之一(即所谓的爱因斯坦空间的存在)是由物理学家称为“全息原理”的猜想推动的。它断言，在爱因斯坦空间中对物理宇宙的任何测量结果都可以通过在宇宙的“远端”(称为其“边界”)附近进行另一组测量来预测。一个特别有趣的情况是当边界空间是三维的情况，在这种情况下，几何理解已经很好地发展了。该项目将吸引研究生和博士后参与其研究活动。

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.