Speaker
Description
This talk is a survey on results concerning the space $T^{<0}(M)$, which we call the Teichmüller space of negatively curved Riemannian metrics on $M$. It is defined as the quotient space of the space of all negatively curved Riemannian metrics on $M$ modulo the space of all isotopies of $M$ that are homotopic to the identity. This space was shown to have highly non-trivial homotopy when $M$ is real hyperbolic by Tom Farrell and Pedro Ontaneda in 2009. Then in 2015, it was shown to be non simply connected in my thesis when $M$ is a suitably chosen Gromov-Thurston manifold (which are examples of negatively curved non-locally symmetric spaces). In 2017, Tom Farrell and myself proved a similar result for $M$ being a suitable complex hyperbolic manifold. In all these results, the dimension of $M$ has to be $4k-2$ for some $k \ge 2$. In this talk, I will explain this project, and talk about the tools we have used so far in unraveling it. I will also mention the cases that are still open in this project.
