Differential equations and the geometry of manifolds

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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.