## Geometry & Topology

### Characterizing the unit ball by its projective automorphism group

Andrew Zimmer

#### Abstract

In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.

#### Article information

Source
Geom. Topol., Volume 20, Number 4 (2016), 2397-2432.

Dates
Accepted: 12 September 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859027

Digital Object Identifier
doi:10.2140/gt.2016.20.2397

Mathematical Reviews number (MathSciNet)
MR3548469

Zentralblatt MATH identifier
1347.53035

#### Citation

Zimmer, Andrew. Characterizing the unit ball by its projective automorphism group. Geom. Topol. 20 (2016), no. 4, 2397--2432. doi:10.2140/gt.2016.20.2397. https://projecteuclid.org/euclid.gt/1510859027

#### References

• M Andersson, M Passare, R Sigurdsson, Complex convexity and analytic functionals, Progress in Mathematics 225, Birkhäuser, Basel (2004)
• Y Benoist, Automorphismes des cônes convexes, Invent. Math. 141 (2000) 149–193
• Y Benoist, Convexes divisibles, I, from: “Algebraic groups and arithmetic”, (S G Dani, G Prasad, editors), Tata Inst. Fund. Res., Mumbai (2004) 339–374
• Y Benoist, A survey on divisible convex sets, from: “Geometry, analysis and topology of discrete groups”, (L Ji, K Liu, L Yang, S-T Yau, editors), Adv. Lect. Math. 6, International Press, Somerville, MA (2008) 1–18
• R Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979) 11–25
• A Cano, J Seade, On discrete groups of automorphisms of $\mathbb{P}\sp 2\sb {\mathbb{C}}$, Geom. Dedicata 168 (2014) 9–60
• B Colbois, P Verovic, Rigidity of Hilbert metrics, Bull. Austral. Math. Soc. 65 (2002) 23–34
• T tom Dieck, Algebraic topology, European Mathematical Society, Zürich (2008)
• L Dubois, Projective metrics and contraction principles for complex cones, J. Lond. Math. Soc. 79 (2009) 719–737
• S Frankel, Complex geometry of convex domains that cover varieties, Acta Math. 163 (1989) 109–149
• W M Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990) 791–845
• W M Goldman, Projective geometry on manifolds, preprint (2009) Available at \setbox0\makeatletter\@url http://www.math.umd.edu/~wmg/pgom.pdf {\unhbox0
• L H örmander, Notions of convexity, Progress in Mathematics 127, Birkhäuser, Boston (1994)
• A V Isaev, S G Krantz, Domains with non-compact automorphism group: a survey, Adv. Math. 146 (1999) 1–38
• K Jo, A rigidity result for domains with a locally strictly convex point, Adv. Geom. 8 (2008) 315–328
• M Kapovich, Convex projective structures on Gromov–Thurston manifolds, Geom. Topol. 11 (2007) 1777–1830
• S Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977) 129–135
• S Kobayashi, T Ochiai, Holomorphic projective structures on compact complex surfaces, Math. Ann. 249 (1980) 75–94
• S G Krantz, The impact of the theorem of Bun Wong and Rosay, Complex Var. Elliptic Equ. 59 (2014) 966–985
• L Marquis, Around groups in Hilbert geometry, from: “Handbook of Hilbert geometry”, (A Papadopoulos, M Troyanov, editors), IRMA Lect. Math. Theor. Phys. 22, Eur. Math. Soc., Zürich (2014) 207–261
• G D Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton Univ. Press (1973)
• J-F Quint, Convexes divisibles (d'après Yves Benoist), from: “Séminaire Bourbaki, Volume 2008/2009”, Astérisque 332, Société Mathématique de France, Paris (2010) Exp. No. 999, 45–73
• J-P Rosay, Sur une caractérisation de la boule parmi les domaines de ${\bf C}\sp{n}$ par son groupe d'automorphismes, Ann. Inst. Fourier $($Grenoble$)$ 29 (1979) 91–97
• E Socié-Méthou, Caractérisation des ellipsoï des par leurs groupes d'automorphismes, Ann. Sci. École Norm. Sup. 35 (2002) 537–548
• S M Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977) 53–68
• B Wong, Characterization of the unit ball in ${\bf C}\sp{n}$ by its automorphism group, Invent. Math. 41 (1977) 253–257
• C Yi, Projective domains with non-compact automorphism groups, I, J. Korean Math. Soc. 45 (2008) 1221–1241
• A M Zimmer, Rigidity of complex convex divisible sets, preprint (2013)
• A M Zimmer, Characterizing polynomial domains by their automorphism group, preprint (2015)
• A M Zimmer, Proper quasi-homogeneous domains in flag manifolds and geometric structures, preprint (2015)