no. 6
Proper quasi-homogeneous domains in flag manifolds and geometric structures
[Domaines quasi-homogènes propres dans les variétés de drapeaux et structures géométriques]
Zimmer, Andrew M.1
1 Louisiana State University Dept. of Mathematics Baton Rouge, LA 70803 (USA)
Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2635-2662.

On étudie dans cet article les domaines dans les variétés de drapeaux dont l’image dans une carte affine est bornée et dont l’action du groupe des automorphismes projectifs est co-compacte. Par contraste avec les nombreux exemples existant dans l’espace projectif réel, on démontre que de nombreuses variétés de drapeaux ne contiennent pas de tels domaines. On établit en outre que dans les cas où l’existence de tels domaines n’est pas exclue, ils sont soumis à une condition de convexité naturelle et possèdent une métrique invariante qui généralise la métrique de Hilbert. Une application de nos résultats fournit des restrictions sur l’application développante de certaines (G,X)-structures.

In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we show that they satisfy a natural convexity condition and have an invariant metric which generalizes the Hilbert metric. As an application we give some restrictions on the developing map for certain (G,X)-structures.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3219
Classification : 22F50, 53C24, 53A20
Keywords: Real projective structures, $(G,X)$-structure, Kobayashi metric, Carathéodory metric, Hilbert metric, projective automorphism group
Mot clés : structures projectives réelles, $(G,X)$-structure, métrique de Kobayashi, métrique de Carathéodory, métrique de Hilbert, automorphismes projectifs

Zimmer, Andrew M. 1

1 Louisiana State University Dept. of Mathematics Baton Rouge, LA 70803 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2018__68_6_2635_0,
     author = {Zimmer, Andrew M.},
     title = {Proper quasi-homogeneous domains in flag manifolds and geometric structures},
     journal = {Annales de l'Institut Fourier},
     pages = {2635--2662},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3219},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3219/}
}
TY  - JOUR
AU  - Zimmer, Andrew M.
TI  - Proper quasi-homogeneous domains in flag manifolds and geometric structures
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2635
EP  - 2662
VL  - 68
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3219/
DO  - 10.5802/aif.3219
LA  - en
ID  - AIF_2018__68_6_2635_0
ER  - 
%0 Journal Article
%A Zimmer, Andrew M.
%T Proper quasi-homogeneous domains in flag manifolds and geometric structures
%J Annales de l'Institut Fourier
%D 2018
%P 2635-2662
%V 68
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3219/
%R 10.5802/aif.3219
%G en
%F AIF_2018__68_6_2635_0
Zimmer, Andrew M. Proper quasi-homogeneous domains in flag manifolds and geometric structures. Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2635-2662. doi : 10.5802/aif.3219. https://aif.centre-mersenne.org/articles/10.5802/aif.3219/

[1] Andersson, Mats; Passare, Mikael; Sigurdsson, Ragnar Complex convexity and analytic functionals, Progress in Mathematics, 225, Birkhäuser, 2004, xii+160 pages | DOI | MR

[2] Ballas, Samuel A.; Danciger, Jeffrey; Lee, Gye-Seon Convex projective structures on non-hyperbolic three-manifolds, Geom. Topol., Volume 22 (2018) no. 3, pp. 1593-1646

[3] Benoist, Yves Automorphismes des cônes convexes, Invent. Math., Volume 141 (2000) no. 1, pp. 149-193 | DOI | MR

[4] Benoist, Yves Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math., Volume 164 (2006) no. 2, pp. 249-278 | DOI | MR

[5] Benoist, Yves A survey on divisible convex sets, Geometry, analysis and topology of discrete groups (Advanced Lectures in Mathematics (ALM)), Volume 6, International Press., 2008, pp. 1-18 | MR | Zbl

[6] Bowen, Rufus Hausdorff dimension of quasicircles, Publ. Math., Inst. Hautes Étud. Sci. (1979) no. 50, pp. 11-25 | MR

[7] Cano, Angel; Seade, José On discrete groups of automorphisms of ℙ ℂ 2 , Geom. Dedicata, Volume 168 (2014), pp. 9-60 | DOI | MR

[8] Dubois, Loïc Projective metrics and contraction principles for complex cones, J. Lond. Math. Soc., Volume 79 (2009) no. 3, pp. 719-737 | DOI | MR

[9] Eberlein, Patrick B. Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, 1996, vii+449 pages | MR

[10] Frankel, Sidney Complex geometry of convex domains that cover varieties, Acta Math., Volume 163 (1989) no. 1-2, pp. 109-149 | DOI | MR

[11] Goldman, William M. Geometric structures on manifolds and varieties of representations, Geometry of group representations (Boulder, CO, 1987) (Contemporary Mathematics), Volume 74, American Mathematical Society, 1988, pp. 169-198 | DOI | MR

[12] Goldman, William M. Convex real projective structures on compact surfaces, J. Differ. Geom., Volume 31 (1990) no. 3, pp. 791-845 http://projecteuclid.org/euclid.jdg/1214444635 | MR

[13] Guéritaud, François; Guichard, Olivier; Kassel, Fanny; Wienhard, Anna Anosov representations and proper actions, Geom. Topol., Volume 21 (2017) no. 1, pp. 485-584 | DOI | MR

[14] Guéritaud, François; Guichard, Olivier; Kassel, Fanny; Wienhard, Anna Compactification of certain Clifford-Klein forms of reductive homogeneous spaces, Mich. Math. J., Volume 66 (2017) no. 1, pp. 49-84 | DOI | MR

[15] Guichard, Olivier; Wienhard, Anna Convex foliated projective structures and the Hitchin component for PSL 4 (R), Duke Math. J., Volume 144 (2008) no. 3, pp. 381-445 | DOI | MR

[16] Guichard, Olivier; Wienhard, Anna Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012) no. 2, pp. 357-438 | DOI | MR

[17] Hörmander, Lars Notions of convexity, Modern Birkhäuser Classics, Birkhäuser, 2007, viii+414 pages (Reprint of the 1994 edition) | MR

[18] Isaev, Alexander V.; Krantz, Steven G. Domains with non-compact automorphism group: a survey, Adv. Math., Volume 146 (1999) no. 1, pp. 1-38 | DOI | MR | Zbl

[19] Johnson, Dennis; Millson, John J. Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, CO, 1984) (Progress in Mathematics), Volume 67, Birkhäuser Boston, 1987, pp. 48-106 | MR

[20] Kapovich, Michael Convex projective structures on Gromov-Thurston manifolds, Geom. Topol., Volume 11 (2007), pp. 1777-1830 | DOI | MR

[21] Kapovich, Michael; Leeb, Bernhard; Porti, Joan Morse actions of discrete groups on symmetric space (2014) (https://arxiv.org/abs/1403.7671)

[22] Kapovich, Michael; Leeb, Bernhard; Porti, Joan A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings (2014) (https://arxiv.org/abs/1411.4176)

[23] Kapovich, Michael; Leeb, Bernhard; Porti, Joan Dynamics on flag manifolds: domains of proper discontinuity and cocompactness, Geom. Topol., Volume 22 (2018) no. 1, pp. 157-234 | DOI | MR

[24] Kobayashi, Shoshichi Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 24 (1977) no. 1, pp. 129-135 | MR | Zbl

[25] Kobayashi, Shoshichi; Ochiai, Takushiro Holomorphic projective structures on compact complex surfaces, Math. Ann., Volume 249 (1980) no. 1, pp. 75-94 | DOI | MR

[26] Koszul, Jean-Louis Déformations de connexions localement plates, Ann. Inst. Fourier, Volume 18 (1968) no. 1, pp. 103-114 | MR

[27] Marquis, Ludovic Around groups in Hilbert geometry, Handbook of Hilbert geometry (IRMA Lectures in Mathematics and Theoretical Physics), Volume 22, European Mathematical Society, 2014, pp. 207-261 | MR

[28] Mostow, G. D. Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, 78, Princeton University Press, 1973, v+195 pages | MR | Zbl

[29] Nagano, Tadashi Transformation groups on compact symmetric spaces, Trans. Am. Math. Soc., Volume 118 (1965), pp. 428-453 | MR

[30] Palais, Richard S. On the existence of slices for actions of non-compact Lie groups, Ann. Math., Volume 73 (1961), pp. 295-323 | MR

[31] Quint, Jean-François Convexes divisibles (d’après Yves Benoist), Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011 (Astérisque), Volume 332, Société Mathématique de France, 2010, p. 45-73, Exp. No. 999 | MR | Zbl

[32] Thurston, William P. Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, 1997, x+311 pages (Edited by Silvio Levy) | MR

[33] Van Limbeek, Wouter; Zimmer, Andrew Rigidity of convex divisible domains in flag manifolds (2015) (https://arxiv.org/abs/1510.04118)

[34] Warner, Garth Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, 188, Springer, 1972, xvi+529 pages | MR | Zbl

[35] Zimmer, Andrew Characterizing the unit ball by its projective automorphism group, Geom. Topol., Volume 20 (2016) no. 4, pp. 2397-2432 | DOI | MR

Cité par Sources :