Differential equations and the geometry of manifolds

摘要：获奖：dms -0503506首席研究员：Jeff A. viaclovsky该奖项支持的第一个项目研究了四维黎曼度量的紧性结果，特别是反自对偶度量和极值kaehler度量。在一定的几何非坍缩假设下，适当的模空间可以通过添加具有广义轨道奇异性的度量而紧化。本课题的长期目标是利用这种紧化来定义新的四流形光滑不变量。第二个项目考虑了度量的保形变形，一定的非线性曲率方程，以及与三维和四维黎曼函数的关系。在这个方向上的一个关键问题是规定里奇张量的特征值的对称函数，推广yamabe问题。几何中反复出现的一个主题是拓扑（空间形状的柔软或弹性版本）与该空间上可能的几何形状之间的关系。同样，在二维球面上熟悉的圆形几何比在球面上推入或拉入小区域获得的任何邻近几何更吸引人，在几何上也更独特，人们喜欢希望大多数低维流形将携带某种最优的几何形状——特别是对称几何形状，或某种总能量测量的最小化。上面描述的项目试图在一些重要的三维和四维情况下发现这种最佳或极端几何形状。

AbstractAward: DMS-0503506Principal Investigator: Jeff A. ViaclovskyThe first project supported by this award deals with compactnessresults for certain classes of Riemannian metrics in dimensionfour, particularly for anti-self-dual metrics and extremalKaehler metrics. With certain geometric noncollapsingassumptions, the appropriate moduli spaces can be compactified byadding metrics with generalized orbifold singularities. Along-term goal of this project is to use this compactification todefine new smooth invariants of four-manifolds. The secondproject considers conformal deformations of metrics, certainfully nonlinear curvature equations, and relations withRiemannian functions in dimensions three and four. A crucialproblem in this direction is to prescribe a symmetric function ofthe eigenvalues of the Ricci tensor, generalizing the Yamabeproblem.A recurring theme in geometry is the relationship betweentopology, a soft or stretchy version of the shape of a space, andthe possible geometries on that space. In the same way that thefamiliar round geometry on the two-dimensional sphere is moreappealing and geometrically distinctive than any of the nearbygeometries obtained by pushing in or pulling small regions on thesphere, one likes to hope that most manifolds of low dimensionswill carry geometries that are somehow optimal - especiallysymmetric geometries, or minimizers for some sort of total energymeasurement. The projects described above seek to discover suchoptimal or extremal geometries in some important three- andfour-dimensional cases.