Diptaishik Choudhury will speak in the geometry seminar at 2pm on 18/10/2022. Diptaishik’s title is “Constant mean curvature surfaces in quasi-Fuchsian manifolds and Hamiltonian paths in cotangent space of Teichmüller space” and his abstract is below.
Quasi-Fuchsian manifolds are a widely studied class of hyperbolic 3-manifolds that are homeomorphic to S × R where S is a closed, oriented, hyperbolic surface. It is known that, in general, these manifolds are not foliated by surfaces of constant mean curvature (CMC) although their ends are. However, a conjecture due to Thurston asks if almost-Fuchsian manifolds admit a foliation by CMC surfaces. Here, almost-Fuchsian manifolds are defined as quasi-Fuchsian manifolds which contain a unique minimal surface with principal curvatures in (−1,1). In this talk, we will discuss proof of the fact that almost-Fuchsian manifolds which are sufficiently close to being Fuchsian are indeed monotonically foliated by surfaces of constant mean curvature. Further, the data of the first and second fundamental forms of the CMC surfaces define a point in the cotangent space of Teichmüller space T*T (S), and the foliation constructed above defines a time-dependent flow. We then show that these paths are the orbits of a Hamiltonian vector field, where the Hamiltonian function is given by the area of the CMC H-surface. This is in the spirit of what was observed by Moncrief (1989) for data of CMC foliations in 3-dimensional Lorentzian spacetime. The work is in collaboration with Filippo Mazzoli and Andrea Seppi.