Complex Analysis and Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract NSF Proposal DMS-0104047 Speaking more technically, the proposal aims to treat four research lines in complex geometry and analysis. We are studying the Grauert tube construction for its rigidity and uniquesness properties for large (maximal) radius or domain, and examining whether this point of view adds to old questions in representation theory and automorphic forms. Second, we will study a Kaehler-Einstein version of Min-Oo's rigidity theorem which we conjecture to hold. We will continue work with X. Gong on Levi flat hypersurfaces with singularities, especially algebraic ones with isolated singularities. Their importance is suggested by recent work of Siu and others proving the conjecture of Camacho on the non-existence of smooth, Levi-flat hypersurfaces in the complex projective plane. Finally we propose to study the rigidity of special classes of Schubert cycles on flag manifolds, following upon the work of the PI's former student M. Walters and, independently, R. Bryant. The proposal addresses several questions in complex analysis and geometry. Most people are familiar with Descartes' analytic geometry from high school: in most respects, this line of investigation is the modern descendent of those early ideas. We will study the relationship between a geometric locus, sometimes defined by equations as in Descartes' original case, and its analytic properties, those properties influenced by the calculus of Newton and Leibniz. In particular we study a complexification of geometric locus, that is, we add "imaginary points" to the geometry, related to the imaginary unit "i", and study the influence of the imaginary points on the real points and their geometry. Another portion of the project seeks to understand the asymptotic, or long range properties of special metric geometries related to the Einstein equations. It is well known that both of these properties can be influential in physical applications, and there is recent work to lead one to hope that both this complexification construction and the asymptotics of Einstein metrics can be used to understand parts of the so-called Maldacena correspondence in theoretical physics.