Non-linear Analysis in Riemannian Geometry

Escobar教授计划继续他在线性和非线性偏微分方程及其在曼几何中的应用方面的研究。他将继续研究紧致黎曼流形上的yamabe方程的解。这些是半线性椭圆方程，在流形有边界的情况下，它们满足非线性边界条件。提出了这些方程的存在性理论的应用，以解决微分几何和相对论中的其他问题。他还提出在二维和三维有边界流形上研究汉密尔顿的里奇流。在二维情况下，这是一个满足非线性边界条件的标量非线性演化方程，而在三维情况下，这是一个具有非线性边界条件的非线性方程组。他将继续研究流形的几何形状和Steklov问题的特征值之间的关系，Steklov问题是研究法向导数与函数成比例的调和函数。我们要研究的几何对象是所谓的黎曼流形。这些空间被赋予了解析结构，就像米制一样，为我们提供了测量长度和角度的方法。研究这些结构的变形是很自然的，以了解在这种扰动下空间中的哪些特性是稳定的。所有这些变形的描述通常由微分方程控制。黎曼流形的曲率张量（黎曼空间的“非欧几里得性”的度量）通常使这样的方程成为非线性的，尽管在物理学中，它们中的大多数是变分性质的。从早期开始，度规的共形变化（度规乘以一个正函数）在曲面理论中起着重要的作用。所提出的方程出现在阿曼度规的共形变形问题和相对论中。Steklov问题出现在数学物理、共形几何、旋量几何、极小曲面、偏微分方程和谐波分析中。几何学和拓扑学都涉及对空间性质的研究。但是，几何关注的是空间的大小、形状和测量等属性，而拓扑学关注的是相对位置和连通性等不那么有形的属性。近年来，哈密顿里奇流已成为流形几何和拓扑学研究的基本工具。

AbstractAward: DMS-0306495Principal Investigator: Jose F. EscobarProfessor Escobar plans to continue his research in linear andnon-linear partial differential equations and its applications toRiemmannian Geometry. He will continue his study of solutions toYamabe equations on compact Riemannian manifolds. These aresemilinear elliptic equations and in the case that the manifoldhas a boundary they satisfy a non-linear boundary condition.Applications of the existence theory for these equations areproposed to solve other problems in differential geometry andrelativity. He also proposes to study Hamilton's Ricci flow ontwo and three dimensional manifols with boundary. In the twodimensional case this is a scalar non-linear evolution equationsatisfying a non-linear boundary condition, while in the threedimensional case is a system of nonlinear equations withnonlinear boundary conditions. He will continue the study of therelation between the geometry of the manifold and the eigenvaluesfor the Steklov problem, which is the study of harmonic functionswhose normal derivative is proportional to the function.The geometric objects that will be studied are the so-calledRiemannian manifolds. These are spaces endowed with analyticalstructures, like the metric which provide us with a way tomeasure lengths and angles. It is natural to study deformationsof these structures to realize what properties in the spaceremain stable under such perturbations. The description of allthese deformations is usually governed by differential equations.The curvature tensor of a Riemmannian manifold ( a measure forthe "non-euclideanness" of a Riemannian space) usually makes suchequations non-linear, although as in physics, most of them are ofvariational nature. From the earliest days, conformal changes ofmetric ( multiplication of the metric by a positive function)have played an importntant role in surface theory. The equationsproposed appear in the problem of conformal deformation of aRiemannian metric and in relativity. The Steklov problem appearsin mathematical physics, conformal geometry, spinor geometry,minimal surfaces, partial differential equations and harmonicanalysis. Geometry as well as topology involves the study ofproperties of the space. But whereas geometry focuses onproperties of space that involves size, shape and measurement,topology concerns itself with the less tangible properties ofrelative position and connectedness. In recent years Hamilton'sRicci flow has become a fundamental tool in the study of geometryand toplogy of manifolds.