Totally geodesic submanifolds of Teichmüller space

Abstract

Let $\mathcal{T}_{g,n}$ and $\mathcal{M}_{g,n}$ denote the Teichmüller and moduli space respectively of genus $g$ Riemann surfaces with $n$ marked points. The Teichmüller metric on these spaces is a natural Finsler metric that quantifies the failure of two different Riemann surfaces to be conformally equivalent. It is equal to the Kobayashi metric [Roy74], and hence reflects the intrinsic complex geometry of these spaces. There is a unique holomorphic and isometric embedding from the hyperbolic plane to $\mathcal{T}_{g,n}$ whose image passes through any two given points. The images of such maps, called Teichmüller disks or complex geodesics, are much studied in relation to the geometry and dynamics of Riemann surfaces and their moduli spaces. A complex submanifold of $\mathcal{T}_{g,n}$ is called totally geodesic if it contains a complex geodesic through any two of its points, and a subvariety of $\mathcal{M}_g$ is called totally geodesic if a component of its preimage in $\mathcal{T}_{g,n}$ is totally geodesic. Totally geodesic submanifolds of dimension $1$ are exactly the complex geodesics. Almost every complex geodesic in $\mathcal{T}_{g,n}$ has dense image in $\mathcal{M}_{g,n}$ [Mas82, Vee82]. We show that higher dimensional totally geodesic submanifolds are much more rigid.