Mathematics

Dr. Saikat Mazumdar
University of British Columbia, Canada

The Yamabe problem, which asks for constant scalar curvature metrics in a conformal class, has proved to be fundamental in the study of nonlinear partial differential equations (PDEs), calculus of variations and differential geometry. In this talk, I will discuss about the higher-order or polyharmonic version of the Yamabe problem: “Given   a compact Riemannian manifold (M, g), does there exists a metric conformal to g with constant Q-curvature” ?

The fascination of Q-curvature stems from its central role in the complex web of ideas where geometry, analysis and theoretical physics meet each other. The behavior of Q-curvature under conformal changes of the metric is governed by certain conformally covariant powers of the Laplacian. The problem of prescribing the Q-curvature in a conformal class then amounts to solving a nonlinear elliptic PDE involving the powers of Laplacian called the GJMS operator. In general the explicit form of this GJMS operator is unknown. This together with a lack of maximum principle makes the problem difficult to tackle. Many questions are still unanswered even for the biharmonic case where the GJMS operator is explicit. In this talk, I will survey some landmark developments and present some of my results in this direction. If time permits, I will also outline some open problems and mention some recent progress.