Differential equations and the geometry of manifolds

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Abstract Award: DMS-0804042 Principal Investigator: Jeff A. ViaclovskyThe first project supported by this award deals with regularity and volume growth properties of critical Riemannian metrics in dimension four, and applications to the compactness of moduli spaces and existence of critical metrics, such as anti-self-dual and extremal Kaehler metrics. With certain geometric noncollapsing assumptions, the appropriate moduli spaces can be compactified by adding metrics orbifold-like singularities. This generalizes results for Einstein metrics to the case of metrics which do not have pointwise Ricci curvature bounds. A long-term goal is to extend the compactness theorem to include the possibility of collapsing, and to find other applications to the differential topology of four-manifolds. The second project deals with orthogonal complex structures, and the relation with subvarieties of twistor spaces. The corresponding equation is conformally invariant, so a natural problem is to find properties of varieties which are invariant under the action of the conformal group. This has applications to understanding the geometry of compact Hermitian manifolds. The third project involves deformation of curvatures, existence of solutions to fully nonlinear curvature equations, and relations with Riemannian functionals on three and four-manifolds. A crucial problem is to conformally deform a metric to prescribe a symmetric function of the eigenvalues of the Ricci tensor (generalizing the Yamabe problem), and to find natural conformally invariant conditions so that a metric can be deformed from a weaker integral pinching condition to a stronger pointwise pinching condition. An important motivation for this research is to understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space which are invariant under continuous stretching or bendings of a space, while the former, geometry, involves understanding distances and is more rigid. For example, the surface of our planet is a sphere, and one measures distances on it by computing arclengths of great circles (the Earth is actually an oblate spheroid, but it is very close to being perfectly spherical). One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. Such a deformation is less appealing that the familiar round Earth, and there are many ways to make this notion very precise in terms of minimizing some sort of total energy measurement. This is directly related to physical principles which say that the state of a physical system will tend towards a final configuration which minimizes the total energy. This idea can be generalized to higher-dimensional objects called manifolds, which are generalized versions of the surface of the our planet. For example, the space that we live in is three-dimensional, and if one includes time, we are in a four-dimensional universe. In order to understand these types of higher-dimensional objects, one attempts to find the best way to measure distances on them which use the least amount of energy, and maximize the symmetries of the space. The projects described above are to define appropriate energies on such spaces in dimensions three and four, and to seek out the important optimal geometries which minimize the total energy.