| Résume | I 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 pairs describing the tangent bundle of the
Teichmueller space of S.
For any such pair, we determine uniquely the pullback metric and the second
fundamental form of the immersion by solving the Gauss - Codazzi equations.
Indeed, solutions of the Gauss-Codazzi equations correspond to critical points
of a suitable “Donaldson -functional” introduced by Gonsalves-Uhlenbeck
(2007), and (in collaboration with M. Lucia an Z. Huang (2022)) we show that
such functional admits a global minimum as its unique critical point.
In addition I shall discuss the asymptotic behavior of those minimizers and
obtain a “convergence” result in terms of the Kodaira map.
For example, in case of genus 2, it is possible to catch at the limit “regular”
CMC 1-mmersions, except in very rare situations which relate to the image, by
the Kodaira map, of the six Weierstrass points of S.
If time permits, I shall mention further progress for higher genus obtained in
collaboration with S. Trapani.
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