Description
Mathematics Colloquium.
Speaker: Saikat Mazumdar.
Affiliation: IIT Bombay.
Date and Time: Wednesday 14 August, 4:00 pm - 5:00 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Yamabe problem and beyond: an interplay of geometry and PDE.
Abstract: Motivated by the theory of compact surfaces, Yamabe wanted to
show that on a given compact Riemannian manifold of any dimension there
always exists a (conformal) metric with constant scalar curvature. It
turns out that solving the Yamabe problem amounts to solving a nonlinear
elliptic partial differential equation (PDE). The solution of the Yamabe
problem by Trudinger, Aubin and Schoen highlighted the local and global
nature of the problem and the unexpected role of the positive mass theorem
of general relativity. In the first part of my talk, I will survey the
Yamabe problem and the related issues of the compactness of solutions.
In the second part of the talk, I will discuss 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 behaviour 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. I will present some of my results
in this direction and mention some recent progress.