Thursday, October 24, 2024 — 12:00 pm — aula M2.2, edificio Matematica
On CMC-immersions of surfaces into Hyperbolic 3-manifolds
Abstract: I shall discuss the so called “moduli space” of Constant Mean Curvature (CMC) c-immersions of a closed surface S (orientable and of genus at least 2) into hyperbolic 3-manifolds.
Interestingly when |c|<1, such space admits a nice parametrization described by elements of the tangent bundle of the Teichmueller space of S.
This is attained by showing that the “constrained” Gauss-Codazzi equations governing the immersion is uniquely solvable, as they admit an associated action functional (the “Donaldson-functional” in Gonsalves-Uhlenbeck (2007)) which has a global minimum as its unique critical point.
On the other hand, (CMC) 1-immersion into the hyperbolic space are particularly relevant in view of their striking analogies with minimal immersions into the Euclidian space. Therefore, I focus on the asymptotic behavior of those minimizers as |c| tends to 1. We show “convergence” to a (CMC) 1-immersion in terms of the Kodaira map.
For example, we show that for genus g=2, it is possible to catch at the limit a “regular” CMC 1-immersions into an hyperbolic 3-manifold, except in very rare situations which relate to the image, under the Kodaira map, of the six Weierstrass points of S.
If time permits, I shall mention f urther progress for higher genus obtained in collaboration with S. Trapani.