The conference "Recent Advances in Geometric Structures" will be held at Hokkaido University on Monday, July 6, 2026, from 10:00 to 17:00.
This conference is organized in conjunction with the visit of Professor Graham Smith as a JSPS Invitational Fellow.
July 6 (Mon)
10:00–11:00 Kentaro Ito "Complex earthquakes and pieces of hyperbolic planes in SL(2,C)"
It is known that earthquake deformations of Fuchsian groups are induced by pleated hyperbolic planes in SL(2,R),
the anti-de Sitter space. We are interested in an extension of this theory to deformations of quasi-Fuchsian groups induced by
surfaces in SL(2,C). At the very beginning of this study, in this talk, we consider pieces of hyperbolic planes in SL(2,C)
inducing complex earthquake deformations of Fuchsian groups.
11:30–12:30 Shinpei Baba "Complex projective structures on Riemann surfaces and the intersections of their holonomy varieties"
A complex projective structure on a Riemann surface has a holonomy representation from its fundamental group in PSL(2, C). Given different diffeomorphic marked Riemann surfaces, we discuss projective structures on those surfaces that share the same holonomy.
14:30–15:30 Graham Smith "On the asymptotic geometry of finite-type $k$-surfaces in $3$-dimensional hyperbolic space"
We define a finite-type $k$-surface to be a complete, immersed surface of finite area and of constant extrinsic curvature equal to $k$. In earlier work, we showed that, for all $k\in]0,1[$, every finite-type $k$ surface has finite many, $N$ say, ends, and that that the space of finite-type $k$-surfaces in $3$-dimensional hyperbolic space is homeomorphic to the space of pointed ramified covers of the Riemann sphere. In this talk we study the asymptotic structure of the ends of such surfaces, showing that each such end is asymptotic to a unique preferred geodesic, which we call the Steiner geodesic. To each finite-type $k$-surface is thus associated $N$ Steiner geodesics, and we show that the vector consisting of all end-points of these geodesics defines a lagrangian immersion from each stratum of the space of finite-type $k$-surfaces into a suitable open subset of the Cartesian product of $2N$ copies of the Riemann sphere.
16:00–17:00 Yoshihiko Matsumoto "CR-invariant energy of Legendrian knots in the Heisenberg group"
We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group, which carries a natural contact structure. This is an analogue of the energy for ordinary knots in Euclidean 3-space due to O'Hara (1991). Whereas O'Hara's energy (more precisely, the one of exponent -2) is invariant under Möbius transformations, our energy for Legendrian knots is invariant under the action of PU(2,1), the group of CR automorphisms of the one-point compactification of the Heisenberg group. I would like to explain carefully how the energy should be defined so as to achieve the PU(2,1)-invariance, and how R-circles, a distinguished class of Legendrian (un)knots, arise as energy minimizers. Time permitting, I will also discuss some open problems. This talk is based on joint work with Jun O'Hara (Chiba University).
Organizer：Shimpei Kobayashi and Jun-ichi Inoguchi (Hokkaido University)