Conformal Geometry and Asymptotically Hyperbolic Manifolds

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AbstractAward: DMS 1308266, Principal Investigator: C. Robin GrahamThe investigator will carry out several research projects studying various aspects of conformal geometry and asymptotically hyperbolic metrics. These include investigation of boundary rigidity for asymptotically hyperbolic metrics, investigation of minimal submanifolds of products of asymptotically hyperbolic spaces with compact manifolds, study of boundary terms for conformally invariant integrals, and study of twisted Dirac operators on asymptotically hyperbolic manifolds and relation to conformally invariant operators on the boundary. The main objectives are to further understanding of these geometries and their relationship. The methods are analytic, geometric, and algebraic, with an intimate connection between these different aspects of the study.This project will focus on the relationship between two different geometric structures: conformal geometry on the one hand and asymptotically hyperbolic geometry on the other. Conformal geometry is the study of properties of space which depend only on angles but not on distances. Hyperbolic geometry involves spaces of negative curvature, in which the analogues of straight lines separate more than in usual flat space. The asymptotic structure of hyperbolic geometry is related to conformal geometry on the lower dimensional boundary at infinity. Several projects will study the relationship between these geometries. Apart from the intrinsic geometric interest, one motivation is the AdS/CFT correspondence in physics, a proposed holographic correspondence for certain physical phenomena. In recent years, the AdS/CFT correspondence has been used to model many strongly coupled physical theories, for instance in condensed matter physics. The proposed activity will further enable the development of human resources through educationally oriented activities of the investigator, including advising, mentoring and teaching graduate students, and curricular development. International cooperation and partnership will be promoted through collaboration between the investigator and researchers in Japan and France. Ties between the mathematics and physics communities will be enhanced. The results will be effectively disseminated through attendance and speaking at meetings and conferences and through posting and publication of articles.