Vector Fields on the Hyperbolic Plane and Surfaces in Half-Pipe Space

Explore a mathematical lecture that delves into the correspondence between vector fields on the hyperbolic plane and surfaces in half-pipe space, presented at the Erwin Schrödinger International Institute's Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications." Learn how Mess's proof of Thurston's earthquake theorem using Anti-de Sitter geometry has influenced the investigation of surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory. Examine the construction of vector fields on the hyperbolic plane and their relationship to surfaces in half-pipe space, which serves as the dual of Minkowski space. Discover the intricacies of extending vector fields from the circle to the hyperbolic plane, with particular focus on infinitesimal earthquakes and harmonic Lagrangian vector fields. Investigate how these vector fields' properties manifest in corresponding surfaces within half-pipe space and their asymptotic boundaries.