Special Metrics in Geometry

Cuny City College, New York NY

Investigators

Abstract

Manifolds are spaces locally resemble Euclidean spaces, and are the main object of research in Geometry, with applications to all areas of mathematics and theoretical physics. The principal investigator's project is centered on one of the most classical problems in Geometry: determining which metric is the best for a given manifold. A metric is a mathematical way of measuring the distance and angles in a manifold. The project's research will require the use of different techniques from several areas of mathematics: differential, algebraic and symplectic geometry, analysis, dynamical systems, and topology. The PI will take advantage of working at The City College of New York to reach underrepresented groups in the sciences, as this institution has an unusually high percentage of minority students. Dr. Santoro's plan of research breaks down into three main projects revolving around various notions of "special metrics" in differential geometry. The special metrics appearing in the first project, about aspects of the classification of ALE scalar-flat surfaces, can be seen as a generalization to gravitational instantons. The PI's second project is the study of a collapsed Dirichlet problem (at infinity) of complete constant mean curvature hypersurfaces in hyperbolic space. Yet another notion of special metrics are the Hermitian-Einstein metrics, a generalization of Kahler-Einstein metrics for vector bundles: in the third project, the PI proposes the development of new notions of stability in Yang-Mills theory for non-holomorphic vector bundles.