Differential equations and the geometry of manifolds

摘要奖：DMS-0503506首席研究员：Jeff A.Viaclovsky该奖项支持的第一个项目涉及四维某些黎曼度量类的紧性结果，特别是反自对偶度量和极端Kaehler度量。在一定的几何非折叠假设下，适当的模空间可以通过添加具有广义orboold奇点的度量来紧致。这个项目的长期目标是利用这种紧化来定义四维流形的新的光滑不变量。第二个方案考虑了度量的共形变形，一定是非线性的曲率方程，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Ricci张量的本征值的对称函数，推广了Yamabe型问题。几何学中反复出现的一个主题是拓扑学(空间形状的软或拉伸版本)与该空间上可能的几何之间的关系。就像二维球体上熟悉的圆形几何比通过推入或拉动球层上的小区域获得的任何邻近几何更吸引人和更独特的几何一样，人们喜欢希望大多数低维流形将带有某种最佳的几何--特别是对称几何，或者是某种总能量测量的最小化。上述项目试图在一些重要的三维和四维情况下发现这样的最佳或极端几何。

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.