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