Non-linear Analysis in Riemannian Geometry

摘要奖：DMS-0306495主要研究者：Jose F. Escobar教授计划继续他的研究在线性和非线性偏微分方程及其应用到黎曼几何。他将继续他的研究解决方案toyamabe方程紧凑黎曼流形。 这类非线性椭圆型方程在流形有边界的情况下满足非线性边界条件，并将其存在性理论应用于微分几何和相对论中的其它问题.他还提出研究具有边界的二维和三维流形上的汉密尔顿里奇流。 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发展方程，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与Steklov问题的特征值之间的关系，Steklov问题是研究其法向导数与函数成比例的调和函数。将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，就像度规，它为我们提供了一种测量长度和角度的方法。 研究这些结构的变形以了解空间中的哪些性质在这种扰动下保持稳定是很自然的。 所有这些变形的描述通常都是由微分方程来控制的。黎曼流形的曲率张量（黎曼空间的“非欧性”的一种度量）通常使这些方程成为非线性的，尽管在物理学中，它们中的大多数都是变分性质的。 从最早的时候起，度规的共形变换（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黎曼度量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、偏微分方程和调和分析中。 几何学和拓扑学都涉及到对空间性质的研究. 但是，几何学关注的是空间的性质，包括大小、形状和测量，而拓扑学关注的是相对位置和连通性等不太有形的性质。近年来，汉密尔顿的Ricci流已成为研究流形几何和拓扑的基本工具。

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.