Linear and Non-Linear Eigenvalues in Geometry

几何中的线性和非线性特征值

• 批准号: 0072164
• 负责人: Robert Strichartz
• 金额: $ 15万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072164&HistoricalAwards=false
• 关键词: Linear; Non; Eigenvalues; Geometry

AbstractAward: DMS-0072164Principal Investigator: Jose F. EscobarProfessor Escobar proposes to work in three different variationalproblems: The first one is on conformal deformation ofmetrics. He proposes to work on the scalar curvature problem onscalar flat manifolds of dimension five or more. In addition, hewill study the prescribed scalar curvature and prescribed meancurvature problem on manifolds with boundary; particularattention will be given to this problem when the manifold is theEuclidean ball and the dimension is three. Earlierinvestigations indicate that the problem in three dimensions isspecial. The second topic he proposes to study is estimates forthe first non-zero Steklov eigenvalue on compact manifolds withboundary. Escobar proposes to study relations between thegeometry of the space and the first non-zero eigenvalue and applythis information to problems in conformal geometry, heat flowproblems, and to the study of eigenvalues of minimalsurfaces. The third topic is to study Einstein metrics onmanifolds with boundary. There are three different kinds ofequations that arise naturally as a variational problem of afunctional introduced by the proposer; they are Einstein metricssatisfying that the boundary is totally geodesic or, moregenerally, that the boundary is umbilic, and Ricci flat metricswith umbilic boundary.The three problems above have their roots in Riemannian geometryas well as in physics. The Steklov problem initially appeared inphysics, then in harmonic analysis, partial differentialequations, conformal geometry, and minimal surfaces. In physics,it describes the temperature of a body where the flux through outthe boundary is proportional to the temperature. We willinvestigate how the geometry of the space influence the firstnon-zero eigenvalue, that is, the smallest nonzero constant ofproportionality. The Einstein equation proposed in this projectis the generalization of the Einstein's equation in boundarylessspaces studied by Hilbert and Einstein in general relativity tothe case of spaces with boundary. The boundary conditions wewill imposed are the natural ones if one studies this problemfrom the point of view of the calculus of variations. The scalarcurvature equations that we will investigate are the averageversion of the Einstein equation on manifolds with boundary.Nowadays they are known as the Yamabe type equations. Theseequations appear in relativity and in other branches of physics.

摘要奖：DMS-0072164首席研究员：Jose F.EScotbar教授建议研究三个不同的变分问题：第一个问题是度量的共形变形。他建议研究五维或更多维的标量平坦流形上的标量曲率问题。此外，他还将研究具有边界的流形上的规定的标量曲率和规定的平均曲率问题；当流形是欧几里德球且维度为三维时，将特别注意这一问题。早期的调查表明，三维问题是特殊的。他建议研究的第二个主题是带边界紧致流形上第一个非零Steklov特征值的估计。埃斯科瓦尔建议研究空间几何与第一非零本征值之间的关系，并将这一信息应用于保角几何问题、热流问题以及最小曲面的特征值的研究。第三个主题是研究有边界流形上的爱因斯坦度量。由作者引入的泛函变分问题自然产生了三种不同类型的方程：爱因斯坦度规和Ricci平坦度规，它们满足边界是全测地线，或者说，边界是脐线的。上述三个问题既有物理根源，也有黎曼几何的根源。Steklov问题最初出现在物理学中，然后出现在调和分析、偏微分方程式、保角几何和极小曲面中。在物理学中，它描述了物体的温度，其中流出边界的流量与温度成正比。我们将研究空间的几何如何影响第一个非零本征值，也就是最小的比例非零常数。本文提出的爱因斯坦方程是希尔伯特和爱因斯坦在广义相对论中研究的无边界空间中爱因斯坦方程在有边界空间中的推广。如果从变分的角度来研究这个问题，我们所施加的边界条件是自然的。我们要研究的标量曲率方程是爱因斯坦方程在有边界流形上的平均变换，现在称为Yamabe型方程。这些方程出现在相对论和物理学的其他分支中。