Non-linear partial differential equations in geometry

在这个项目中，主要研究人员将研究几个高阶非线性偏微分方程，出现在共形几何和柯西-黎曼（CR）几何。这些方程描述了对新结构的曲率进行共形改变的度量（如在共形几何中）或接触形式（如在CR几何中）的影响。特别令人感兴趣的是方程，规定高阶曲率不变量，控制全局几何的高维流形。该项目旨在开发新的工具来解决这些方程，并描述解决方案的行为。在处理这样的方程的一个关键困难是最大值原理的缺乏，和主要的研究人员提出，以弥补其缺乏使用积分不等式作为主要的工作工具。预计该研究将产生著名的混合体积不等式扩展到更一般的领域在欧几里德空间，以及尖锐的Sobolev不等式在CR几何。该项目的目标是提供新的见解，并创造新的工具，为研究几何偏微分方程的理论，一个主题，已演变成一个基本的工具箱在许多领域的数学，应用数学和工程学，更不用说数学物理学了。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性）是由物理学家称之为“全息原理”的猜想所激发的。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。一个特别令人感兴趣的情况是当边界空间是三维时的情况，在这种情况下，几何理解已经得到了很好的发展。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.