Mathematical Sciences: Differential Structures

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Eugenio Calabi will continue studying obstructions to the existence of a metric with constant curvature on a compact Riemann surface with finitely many prescribed singularities; Bernstein problems associated with convex surfaces having extremal, affinely invariant area; relations between the singularities of minimal subvarieties and their Morse index. Hermann Gluck will continue his work on minimizing cycles in homogeneous spaces; the Blaschke problem and fibrations of spheres by great spheres; and the topology of the space of Morse- Smale vector fields on surfaces. Wolfgang Ziller will carry out research into Einstein metrics, calibrations and the rigidity of symmetric spaces. Calabi's research is in the area of geometric variational problems. The problems he will study concern extremal Kahler metrics, the affine Plateau problem and a variational theory of geodesics and minimal subvarieties. Gluck will be continuing the search for the subvarieties of Lie groups and homogeneous spaces which are volume minimizing in their homology classes. He will also work on the famous Blaschke conjecture that every Blaschke manifold is isometric to a round sphere or projective space. His work on Morse-Smale vector fields is concerned with finding the homotopy type of the components of the set of such fields on the two-sphere. Ziller's work on Einstein metrics is concerned with finding information about manifolds which admit such metrics, that is, having constant Ricci curvature. His work on symmetric spaces involves making local changes on the metric whilst preserving the bounds on the curvature. Such changes result in metrics isometric to the original metric. He will also use the method of calibrations to investigate which subgrassmannians have minimal volume in their homology class.