# 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. Indeed, for any such element we shall see how to determine uniquely the pullback metric and the second fundamental form of the immersion by solving the “constrained” Gauss - Codazzi equations. This is attained by showing that the associated action functional ( known as the “Donaldson -functional” in Gonsalves-Uhlenbeck (2007)) admits a global minimum as its unique critical point. In addition I shall discuss the asymptotic behavior of those minimizers and obtain “convergence” to a (CMC) 1-immersion in terms of the Kodaira map. Please note that (CMC) 1-immersion into the hyperbolic space are particularly relevant in hyperbolic geometry in view of their analogies with minimal immersions into the Euclidean space. For example, we show that for genus 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 further progress for higher genus obtained in collaboration with S. Trapani.

**Bio:**

Gabriella Tarantello is an Italian mathematician specializing in partial differential equations, differential geometry, and gauge theory. She is full professor in the Department of Mathematics at the University of Roma Tor Vergata. Her roots are in Abruzzo, where she completed the undergraduate studies in Mathematics at the University of L'Aquila in 1982. She then pursued further education at the Courant Institute of Mathematical Sciences, obtaining the "Master of Arts and Sciences in Mathematics" and subsequently, in 1986, the Ph.D. Her dissertation, titled "Some Results on the Minimal Period Problem for Nonlinear Vibrating Strings and Hamiltonian Systems, and on the Number of Solutions for Semilinear Elliptic Equations", was supervised by Louis Nirenberg. Following a postdoctoral research period at the Institute for Advanced Study and a visiting assistant professorship at the University of California, Berkeley, she joined the faculty at Carnegie Mellon University. She returned to Italy as an associate professor at Tor Vergata in 1993, becoming full professor two years later. Her network of international collaborations is extensive, and her scientific activities include many visits to prestigious institutions. She has lectured at several schools on nonlinear PDEs in Geometry and Physics and has mentored many students. Gabriella Tarantello is the author of over fifty research papers and of the renowned monograph "Self-dual Gauge Field Vortices: An Analytical Approach".

## 13:30 - 14:30_Meeting with Students_ "Q&A Session with the guest: Career in Mathematics in Italy".

Refreshments | 15:30—16:30 | Common Room Ground Floor