# Minimizing immersions of surfaces in hyperbolic 3-manifolds

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Francesco Bonsante

created by raffero on 29 Dec 2020

modified on 15 Jan 2021

27 jan 2021
-- 17:00

Differential Geometry Seminar Torino (online)

**Abstract.**

Trapani and Valle proposed to study the $L^1$ holomorphic energy of diffeomorphisms between Riemannian surfaces.

This is defined as the $L^1$-norm of the $(1,0)$-part of the differential of the map. They proved that if the domain and the target are surfaces of negative curvature, any homotopy class of diffeomorphisms contains a unique minimizer for the functional.

In a recent work with Gabriele Mondello and Jean-Marc Schlenker we tried to generalize the functional in the setting where the domain is a hyperbolic surface and the target a hyperbolic 3-manifold.

The functional here is the $L^1$-Shatten energy, which in fact coincides with the $L^1$-holomorphic energy in the 2-dimensional case.

More concretely we considered the space of equivariant maps of the universal covering of a fixed surface of genus $g$ into the hyperbolic space, and studied maps which minimize the $L^1$-Shatten energy on fibers of the monodromy map. We proved that the space of such minimizing maps is naturally a complex manifold of dimension $6g-6$, where $g$ is the genus of the surface, so that the monodromy map realize a holomorphic embedding onto some open subset of the PSL$_2$(C)-character variety containing the Fuchsian locus.

In the talk I will describe the main results of this joint work.

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