Circle packing and Teichmüller space. (English) Zbl 1208.53053
Papadopoulos, Athanase (ed.), Handbook of Teichmüller theory. Volume II. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-055-5/hbk). IRMA Lectures in Mathematics and Theoretical Physics 13, 509-531 (2009).
From the author’s introduction: We are interested in a circle packing on a Riemann surface endowed with a projective structure, which will be a particular configuration of circles such that all complementary regions are curvilinear triangles. It enjoys both rigid and flexible properties in connection with Teichmüller spaces. The main purpose of this chapter is to discuss such interesting properties from geometric viewpoints.
The rigidity we discuss here has been worked out by P. Koebe [Ber. Verh. Sächs. Akad. Leipzig 88, 141–164 (1936; Zbl 0017.21701, JFM 62.1217.04)], E. M. Andreev [Math. USSR, Sb. 12(1970), 255–259 (1971); translation from Mat. Sb., N. Ser. 83(125), 256–260 (1970; Zbl 0203.54904)] and W. P. Thurston [The geometry and topology of 3-manifolds, Lect. Notes, Princeton University (1979)] for realization of a circle packing on a constant curvature surface. More specifically, we describe in a rather uniform way how a combinatorial adjacency data of circles determines uniquely a constant curvature surface which supports a geometric packing with prescribed data.
This rigidity motivated R. Brooks [Duke Math. J. 52, 1009–1024 (1985; Zbl 0587.58060); Invent. Math. 86, 461–469 (1986; Zbl 0578.30037)] to analyze the flexibility phenomenon when curvilinear quadrilateral complementary regions are allowed. He succeeded to parametrize the deformation space in terms of continued fractional type numerical invariants, and deduced the density of packable constant curvature surfaces in Teichmüller space. We discuss Brooks’ idea briefly, and preprint how his parameters work through quasi-conformal deformation theory.
On the other hand, extending the problem Koebe-Andreev-Thurston settled on constant curvature surfaces, one may ask what the set of projective Riemann surfaces supporting a circle packing with a common combinatorial data looks like. It leads us to analyze the flexibility of the object in question. Following [S. Kojima, S. Mizushima and S. P. Tan, J. Differ. Geom. 63, No. 3, 349–397 (2003; Zbl 1075.52509); Pac. J. Math. 225, No. 2, 287–300 (2006; Zbl 1130.52009); in: Minsky, Yair (ed.) et al., Spaces of Kleinian groups. Proceedings of the programme ‘Spaces of Kleinian groups and hyperbolic 3-manifolds’, Cambridge, UK, July 21–August 15, 2003. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 329, 337–353 (2006; Zbl 1102.52009)], we present here a construction of the moduli space of pairs of such projective Riemann surfaces with circle packings, and present its basic properties. In particular, we discuss our belief that the moduli space provides a sort of uniformization in terms of circle packing. We state this as a conjecture in more explicit form, and report some progress towards it.
For the entire collection see [Zbl 1158.30001].
The rigidity we discuss here has been worked out by P. Koebe [Ber. Verh. Sächs. Akad. Leipzig 88, 141–164 (1936; Zbl 0017.21701, JFM 62.1217.04)], E. M. Andreev [Math. USSR, Sb. 12(1970), 255–259 (1971); translation from Mat. Sb., N. Ser. 83(125), 256–260 (1970; Zbl 0203.54904)] and W. P. Thurston [The geometry and topology of 3-manifolds, Lect. Notes, Princeton University (1979)] for realization of a circle packing on a constant curvature surface. More specifically, we describe in a rather uniform way how a combinatorial adjacency data of circles determines uniquely a constant curvature surface which supports a geometric packing with prescribed data.
This rigidity motivated R. Brooks [Duke Math. J. 52, 1009–1024 (1985; Zbl 0587.58060); Invent. Math. 86, 461–469 (1986; Zbl 0578.30037)] to analyze the flexibility phenomenon when curvilinear quadrilateral complementary regions are allowed. He succeeded to parametrize the deformation space in terms of continued fractional type numerical invariants, and deduced the density of packable constant curvature surfaces in Teichmüller space. We discuss Brooks’ idea briefly, and preprint how his parameters work through quasi-conformal deformation theory.
On the other hand, extending the problem Koebe-Andreev-Thurston settled on constant curvature surfaces, one may ask what the set of projective Riemann surfaces supporting a circle packing with a common combinatorial data looks like. It leads us to analyze the flexibility of the object in question. Following [S. Kojima, S. Mizushima and S. P. Tan, J. Differ. Geom. 63, No. 3, 349–397 (2003; Zbl 1075.52509); Pac. J. Math. 225, No. 2, 287–300 (2006; Zbl 1130.52009); in: Minsky, Yair (ed.) et al., Spaces of Kleinian groups. Proceedings of the programme ‘Spaces of Kleinian groups and hyperbolic 3-manifolds’, Cambridge, UK, July 21–August 15, 2003. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 329, 337–353 (2006; Zbl 1102.52009)], we present here a construction of the moduli space of pairs of such projective Riemann surfaces with circle packings, and present its basic properties. In particular, we discuss our belief that the moduli space provides a sort of uniformization in terms of circle packing. We state this as a conjecture in more explicit form, and report some progress towards it.
For the entire collection see [Zbl 1158.30001].
MSC:
| 53C24 | Rigidity results |
| 53A20 | Projective differential geometry |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 30F60 | Teichmüller theory for Riemann surfaces |
| 52C26 | Circle packings and discrete conformal geometry |
