Principal Investigator: Jeff A. Viaclovsky
The first project supported by this award deals with compactness results for certain classes of Riemannian metrics in dimension four, particularly for anti-self-dual metrics and extremal Kaehler metrics. With certain geometric noncollapsing assumptions, the appropriate moduli spaces can be compactified by adding metrics with generalized orbifold singularities. A long-term goal of this project is to use this compactification to define new smooth invariants of four-manifolds. The second project considers conformal deformations of metrics, certain fully nonlinear curvature equations, and relations with Riemannian functions in dimensions three and four. A crucial problem in this direction is to prescribe a symmetric function of the eigenvalues of the Ricci tensor, generalizing the Yamabe problem.
A recurring theme in geometry is the relationship between topology, a soft or stretchy version of the shape of a space, and the possible geometries on that space. In the same way that the familiar round geometry on the two-dimensional sphere is more appealing and geometrically distinctive than any of the nearby geometries obtained by pushing in or pulling small regions on the sphere, one likes to hope that most manifolds of low dimensions will carry geometries that are somehow optimal - especially symmetric geometries, or minimizers for some sort of total energy measurement. The projects described above seek to discover such optimal or extremal geometries in some important three- and four-dimensional cases.
