Minimizing immersions of surfaces in hyperbolic 3-manifolds

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.

differential geometry

Audience: researchers in the topic

Differential Geometry Seminar Torino

Series comments: This is a hybrid seminar organized by the Differential Geometry groups of Università and Politecnico di Torino.

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