Linear and Non-Linear Eigenvalues in Geometry

摘要/ abstract摘要：dms -0072164首席研究员：Jose F. Escobar教授提出了三个不同的变分问题：第一个是度量的保形变形。他建议研究五维或更多维标量平面流形上的标量曲率问题。此外，他还将研究有边界流形上的规定标量曲率和规定曲率问题；当流形为欧几里得球且维数为3时，将特别注意这个问题。早先的研究表明，三维空间的问题是特殊的。他提出研究的第二个主题是有边界的紧流形上第一个非零Steklov特征值的估计。Escobar建议研究空间几何与第一非零特征值之间的关系，并将此信息应用于共形几何问题，热流问题以及最小曲面特征值的研究。第三个课题是研究有边界流形上的爱因斯坦度量。有三种不同的方程自然地出现作为一个变分问题的泛函由提议者引入；它们是爱因斯坦度规，满足边界是完全测地线的，或者更一般地说，满足边界是脐带形的，而里奇是具有脐带形边界的平坦度规。上述三个问题在黎曼几何和物理学中都有根源。Steklov问题最初出现在物理学中，然后出现在谐波分析、偏微分方程、共形几何和最小曲面中。在物理学中，它描述了一个物体的温度，其中通过边界的通量与温度成正比。我们将研究空间的几何形状如何影响第一个非零特征值，即最小的非零比例常数。本文提出的爱因斯坦方程将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的爱因斯坦方程推广到有边界空间的情况。如果从变分学的观点来研究这个问题，我们将施加的边界条件是自然的。我们将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演化。现在它们被称为Yamabe型方程。这些方程出现在相对论和物理学的其他分支中。

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.