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July 2020 Totally geodesic submanifolds of Teichmüller space
Alex Wright
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J. Differential Geom. 115(3): 565-575 (July 2020). DOI: 10.4310/jdg/1594260019


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.


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Alex Wright. "Totally geodesic submanifolds of Teichmüller space." J. Differential Geom. 115 (3) 565 - 575, July 2020.


Received: 10 February 2017; Published: July 2020
First available in Project Euclid: 9 July 2020

zbMATH: 07225031
MathSciNet: MR4120819
Digital Object Identifier: 10.4310/jdg/1594260019

Rights: Copyright © 2020 Lehigh University


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Vol.115 • No. 3 • July 2020
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