Event calendar file
School of Mathematics Seminars and Lectures
Some rigidity theorems for simply connected negatively curved manifolds.
by Prof. Kingshook Biswas (RKMVU, Belur)
Wednesday, July 30, 2014
from
to
(Asia/Kolkata)
at Colaba Campus ( AG-77 )
at Colaba Campus ( AG-77 )
| Description |
The boundary at infinity has proved to be a very useful tool in the study of rigidity properties of negatively curved spaces. In particular for CAT(-1) spaces there is a canonical cross-ratio function defined on quadruples of points on the boundary, and there are notions of Moebius and conformal mappings between boundaries of CAT(-1) spaces. A motivating open question is whether any Moebius map between two boundaries extends to an isometry inside. This is proved by Bourdon to be true if the domain is the boundary of a rank one symmetric space of noncompact type. For a conformal map f between boundaries of spaces X,Y, we define a function S(f) on the space of geodesics of X, called the integrated Schwarzian of f, which measures the deviation of f from being Moebius. Like the classical Schwarzian derivative, the integrated Schwarzian satisfies a cocycle identity, and vanishes if and only if f is Moebius. We use the integrated Schwarzian to study simply connected negatively curved manifolds. We show that a 1-parameter family of compactly supported metric deformations is an isometric deformation if all the boundary maps in the family are Moebius. We also prove, under certain hypotheses, that any small enough compactly supported metric deformation is isometric if the boundary map is Moebius. |
