Canonical metrics and stability

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Abstract Award: DMS-0514003 Principal Investigator: Jacob Sturm The main themes of this proposal - critical exponents of holomorphic functions, the Kaehler-Ricci flow, canonical metrics and stability of algebraic varieties, are well established in classical analysis and differential geometry, with fundamental results dating back to the nineteenth century. The principal investigator will continue his research with D.H. Phong in these areas: The first project will use techniques developed in earlier work to understand the critical exponent set (i.e., the set of log canonical thresholds) in n-dimensions. Of particular interest are the applications of this structure to problems in algebraic geometry. The second project concerns the Kaehler-Ricci flow, and addresses the convergence properties of this flow and the question of when the positivity of the Ricci curvature is preserved under the flow. The third project is motivated by the basic question: When can a Kaehler metric on a compact complex manifold be deformed into one of constant scalar curvature? It is believed that the key to this problem lies in the relationship between the finite geometry of the general linear group and the infinite geometry of the space of Kaehler metrics. Techniques from pluri-potential theory, developed in earlier work, will be applied to elucidate this relationship. The laws of nature are written in the language of mathematics, and conversely, much of the research in modern mathematics is spawned by fundamental discoveries in science. Prime examples of this interplay are seen in the theory of general relativity and in quantum field theory, which were derived with the help of the tensor calculus of differential geometry. On the other hand, the equations which govern our universe (which include the Einstein equation and the Yang-Mills equation) have a beautiful structure which, for some mysterious reason, has helped elucidate fundamental questions in abstract mathematics. The problems outlined in this proposal fall into this very productive and beautiful interplay between the understanding of our physical universe and the abstract realm of mathematics.