Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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Minimizing immersions of surfaces in hyperbolic 3-manifolds

Francesco Bonsante

created by raffero on 29 Dec 2020
modified on 15 Jan 2021

27 jan 2021 -- 17:00

Differential Geometry Seminar Torino (online)


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.


The link of the virtual room will be sent one day before the talk to all registered participants. To register, please send an e-mail to

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