Recent Advances in Geometric Structures¶
Overview¶
The conference Recent Advances in Geometric Structures will be held at Hokkaido University on 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.
Registration¶
Registration Form
Registration is open until July 3, 2026.
Place and how to get there¶
Hokkaido University, School of Science Building 4, Room 501
Fourth floor
Speakers¶
- Graham Smith
- Kentaro Ito
- Yoshihiko Matsumoto
- Shinpei Baba
Schedule¶
July 6, 2026, Monday¶
| Time | Speaker | Title |
|---|---|---|
| 10:00--11:00 | Kentaro Ito | Complex earthquakes and pieces of hyperbolic planes in \(\mathrm{SL}(2,\mathbb{C})\) |
| 11:30--12:30 | Shinpei Baba | Bending Teichmüller spaces and character varieties |
| 14:30--15:30 | Graham Smith | On the asymptotic geometry of finite-type \(k\)-surfaces in \(3\)-dimensional hyperbolic space |
| 16:00--17:00 | Yoshihiko Matsumoto | CR-invariant energy of Legendrian knots in the Heisenberg group |
Abstracts¶
Kentaro Ito¶
Complex earthquakes and pieces of hyperbolic planes in \(\mathrm{SL}(2,\mathbb{C})\)
It is known that earthquake deformations of Fuchsian groups are induced by pleated hyperbolic planes in \(\mathrm{SL}(2,\mathbb{R})\), the anti-de Sitter space. We are interested in an extension of this theory to deformations of quasi-Fuchsian groups induced by surfaces in \(\mathrm{SL}(2,\mathbb{C})\). At the very beginning of this study, in this talk, we consider pieces of hyperbolic planes in \(\mathrm{SL}(2,\mathbb{C})\) inducing complex earthquake deformations of Fuchsian groups.
Shinpei Baba¶
Bending Teichmüller spaces and character varieties
Let S be a closed oriented surface of genus at least two. The Teichmüller space of S can be identified with the space of discrete faithful representations from the fundamental group of S into PSL(2, R). Given a simple closed curve on S with positive weight (or more generally, a measured lamination), we can “bend” the representation along the curve by an angle equal to the weight, and obtain a representation of the surface group into PSL(2, C). This bending deformation induces a mapping from the Teichmüller space into the space of representations of the surface group into PSL(2, C). We discuss some interesting properties of this mapping.
If time permits, we also discuss a complexification of this mapping, constructed geometrically.
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.
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 \(\mathrm{PU}(2,1)\), 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 \(\mathrm{PU}(2,1)\)-invariance, and how R-circles, a distinguished class of Legendrian unknots and 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.
Organizers¶
Shimpei Kobayashi and Jun-ichi Inoguchi
Hokkaido University