Extension of homeomorphisms and vector fields of the circle: From Anti-de Sitter to Minkowski geometry.

Geometry Topology Seminar
Monday, May 1, 2023 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Farid Diaf – Université Grenoble Alpes – diaffarid97@gmail.com
Alex Nolte

In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory.

In this spirit, we investigate the problem of the existence of vector fields giving infinitesimal earthquakes on the hyperbolic plane, using the so-called Half-pipe geometry which is the dual of Minkowski geometry in a suitable sense. In particular, we recover Gardiner's theorem, which states that any Zygmund vector field on the circle can be represented as an infinitesimal earthquake. Our findings suggest a connection between vector fields on the hyperbolic plane and surfaces in 3-dimensional Half-pipe space, which may be suggestive of a bigger picture.