Categoria:
Seminari di Algebra e Geometria
Data e ora inizio evento:
Data e ora fine evento:
Aula:
Sala di Consiglio
Sede:
Dipartimento di Matematica, Sapienza Università di Roma
Speaker:
Gabriella Tarantello (Università di Roma "Tor Vergata")
We discuss a parametrization for the moduli space of Constant Mean Curvature (CMC)immersions of a closed surface S (orientable and of genus at least 2) into hyperbolic 3-manifolds by elements of the tangent bundle of the Teichmüller space of S. Namely, by pairs formed of a given conformal structure X on S and a Dolbeault cohomology class of (0,1)-forms valued in the holomorphic tangent bundle of X. For any such pair, we determine uniquely the pullback metric and the second fundamental form of the immersion by solving the Gauss-Codazzi equations. The Gauss-Codazzi equations can be viewed as Hitchin's self-duality equations for a suitable nilpotent SL(2;C)-Higgs bundle, and have been handled in this way in case of minimal immersions. However, they correspond also to the Euler-Lagrange equation of a suitable Donaldson functional [see Gonsalves-Uhlenbeck (2007)] and their unique solvability is attained [in collaboration with M. Lucia an Z. Huang (2022)] by showing that such functional admits a global minimum as its unique critical point. Eventually, we can extend such a uniqueness result to more general situations previously treated via the Higgs-bundle approach, including minimal Lagrangian immersions. In addition, we are able to analyze the asymptotic behavior of those minimizers along a whole ray of cohomology classes and obtain "convergence" in terms of the Kodaira map. For example in case of genus 2, we are able to catch at the limit "regular" CMC 1-immersions, except in the rare situation where the projective representative of given cohomology class belongs to the image, through the Kodaira map, of the six Weierstrass points of S. If time permits, we shall mention further recent progress for higher genus obtained in collaboration with S. Trapani.