Maximal representations of uniform complex hyperbolic lattices

Abstract

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset\mathrm{SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $G$. We prove that necessarily $G=\mathrm{SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $\rho$-equivariant map from the complex hyperbolic $n$-space to the symmetric space associated to $\mathrm{SU}(p,q)$. This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of $\mathrm{SU}(p,q)$, the representation $\rho$ extends to a representation of $\mathrm{SU}(n,1)$ in $\mathrm{SU}(p,q)$.

Note: To view the article, click on the URL link for the DOI number.

  • [BMQ] Go to document F. A. Bogomolov and M. L. McQuillan, Rational curves on foliated varieties, 2001.
    @MISC{BMQ,
      author = {Bogomolov, F. A. and McQuillan, M. L.},
      note = {IHES preprint},
      title = {Rational curves on foliated varieties},
      year = {2001},
      URL = {preprints.ihes.fr/M01/M01-07.ps.gz},
      }
  • [BGPG1] Go to document S. B. Bradlow, O. Garc’ia-Prada, and P. B. Gothen, "Surface group representations and ${ U}(p,q)$-Higgs bundles," J. Differential Geom., vol. 64, iss. 1, pp. 111-170, 2003.
    @ARTICLE{BGPG1, mrkey = {2015045},
      number = {1},
      issn = {0022-040X},
      author = {Bradlow, Steven B. and Garc{\'ı}a-Prada, Oscar and Gothen, Peter B.},
      mrclass = {53D30 (57R19)},
      url = {http://projecteuclid.org/euclid.jdg/1090426889},
      journal = {J. Differential Geom.},
      zblnumber = {1070.53054},
      volume = {64},
      mrnumber = {2015045},
      fjournal = {Journal of Differential Geometry},
      mrreviewer = {Ignasi Mundet-Riera},
      coden = {JDGEAS},
      title = {Surface group representations and {${\rm U}(p,q)$}-{H}iggs bundles},
      year = {2003},
      pages = {111--170},
      }
  • [BGPG2] Go to document S. B. Bradlow, O. Garc’ia-Prada, and P. B. Gothen, "Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces," Geom. Dedicata, vol. 122, pp. 185-213, 2006.
    @ARTICLE{BGPG2, mrkey = {2295550},
      issn = {0046-5755},
      author = {Bradlow, Steven B. and Garc{\'ı}a-Prada, Oscar and Gothen, Peter B.},
      mrclass = {14D20 (53D30 58D29)},
      doi = {10.1007/s10711-007-9127-y},
      journal = {Geom. Dedicata},
      zblnumber = {1132.14029},
      volume = {122},
      mrnumber = {2295550},
      fjournal = {Geometriae Dedicata},
      mrreviewer = {Anna Wienhard},
      coden = {GEMDAT},
      title = {Maximal surface group representations in isometry groups of classical {H}ermitian symmetric spaces},
      year = {2006},
      pages = {185--213},
      }
  • [BH] Go to document M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, New York: Springer-Verlag, 1999, vol. 319.
    @BOOK{BH, mrkey = {1744486},
      author = {Bridson, Martin R. and Haefliger, Andr{é}},
      mrclass = {53C23 (20F65 53C70 57M07)},
      series = {Grundl. Math. Wissen.},
      address = {New York},
      isbn = {3-540-64324-9},
      publisher = {Springer-Verlag},
      doi = {10.1007/978-3-662-12494-9},
      zblnumber = {0988.53001},
      volume = {319},
      mrnumber = {1744486},
      mrreviewer = {Athanase Papadopoulos},
      title = {Metric Spaces of Non-positive Curvature},
      year = {1999},
      pages = {xxii+643},
      }
  • [BI07] Go to document M. Burger and A. Iozzi, "Bounded differential forms, generalized Milnor-Wood inequality and an application to deformation rigidity," Geom. Dedicata, vol. 125, pp. 1-23, 2007.
    @ARTICLE{BI07, mrkey = {2322535},
      issn = {0046-5755},
      author = {Burger, Marc and Iozzi, Alessandra},
      mrclass = {53C24 (22E41)},
      doi = {10.1007/s10711-006-9108-6},
      journal = {Geom. Dedicata},
      zblnumber = {1134.53020},
      volume = {125},
      mrnumber = {2322535},
      fjournal = {Geometriae Dedicata},
      coden = {GEMDAT},
      title = {Bounded differential forms, generalized {M}ilnor-{W}ood inequality and an application to deformation rigidity},
      year = {2007},
      pages = {1--23},
      }
  • [BI08] Go to document M. Burger and A. Iozzi, "A measurable Cartan theorem and applications to deformation rigidity in complex hyperbolic geometry," Pure Appl. Math. Q., vol. 4, iss. 1, Special Issue: In honor of Grigory Margulis. Part 2, pp. 181-202, 2008.
    @ARTICLE{BI08, mrkey = {2406001},
      number = {1, Special Issue: In honor of Grigory Margulis. Part 2},
      issn = {1558-8599},
      author = {Burger, Marc and Iozzi, Alessandra},
      mrclass = {22E40 (30F45 32Q45)},
      doi = {10.4310/PAMQ.2008.v4.n1.a8},
      journal = {Pure Appl. Math. Q.},
      zblnumber = {1145.32013},
      volume = {4},
      mrnumber = {2406001},
      fjournal = {Pure and Applied Mathematics Quarterly},
      mrreviewer = {John R. Parker},
      title = {A measurable {C}artan theorem and applications to deformation rigidity in complex hyperbolic geometry},
      year = {2008},
      pages = {181--202},
      }
  • [BIW09] Go to document M. Burger, A. Iozzi, and A. Wienhard, "Tight homomorphisms and Hermitian symmetric spaces," Geom. Funct. Anal., vol. 19, iss. 3, pp. 678-721, 2009.
    @ARTICLE{BIW09, mrkey = {2563767},
      number = {3},
      issn = {1016-443X},
      author = {Burger, Marc and Iozzi, Alessandra and Wienhard, Anna},
      mrclass = {53C24 (32M15 53C35)},
      doi = {10.1007/s00039-009-0020-8},
      journal = {Geom. Funct. Anal.},
      zblnumber = {1188.53050},
      volume = {19},
      mrnumber = {2563767},
      fjournal = {Geometric and Functional Analysis},
      mrreviewer = {Michelle Bucher-Karlsson},
      coden = {GFANFB},
      title = {Tight homomorphisms and {H}ermitian symmetric spaces},
      year = {2009},
      pages = {678--721},
      }
  • [BIW] Go to document M. Burger, A. Iozzi, and A. Wienhard, "Surface group representations with maximal Toledo invariant," Ann. of Math., vol. 172, iss. 1, pp. 517-566, 2010.
    @ARTICLE{BIW, mrkey = {2680425},
      number = {1},
      issn = {0003-486X},
      author = {Burger, Marc and Iozzi, Alessandra and Wienhard, Anna},
      mrclass = {22E41 (20F67 57M07)},
      doi = {10.4007/annals.2010.172.517},
      journal = {Ann. of Math.},
      zblnumber = {1208.32014},
      volume = {172},
      mrnumber = {2680425},
      fjournal = {Annals of Mathematics. Second Series},
      coden = {ANMAAH},
      title = {Surface group representations with maximal {T}oledo invariant},
      year = {2010},
      pages = {517--566},
      }
  • [CW] Go to document J. -G. Cao and P. Wong, "Finsler geometry of projectivized vector bundles," J. Math. Kyoto Univ., vol. 43, iss. 2, pp. 369-410, 2003.
    @ARTICLE{CW, mrkey = {2051030},
      number = {2},
      issn = {0023-608X},
      author = {Cao, J.-G. and Wong, Pit-Mann},
      mrclass = {53C60},
      journal = {J. Math. Kyoto Univ.},
      zblnumber = {1086.53092},
      volume = {43},
      mrnumber = {2051030},
      fjournal = {Journal of Mathematics of Kyoto University},
      mrreviewer = {Min Ru},
      coden = {JMKYAZ},
      title = {Finsler geometry of projectivized vector bundles},
      year = {2003},
      pages = {369--410},
      url = {http://projecteuclid.org/euclid.kjm/1250283732},
      }
  • [CorletteFlatGBundles] Go to document K. Corlette, "Flat $G$-bundles with canonical metrics," J. Differential Geom., vol. 28, iss. 3, pp. 361-382, 1988.
    @ARTICLE{CorletteFlatGBundles, mrkey = {0965220},
      number = {3},
      issn = {0022-040X},
      author = {Corlette, Kevin},
      mrclass = {58E20 (32L99 53C10)},
      url = {http://projecteuclid.org/euclid.jdg/1214442469},
      journal = {J. Differential Geom.},
      zblnumber = {0676.58007},
      volume = {28},
      mrnumber = {0965220},
      fjournal = {Journal of Differential Geometry},
      mrreviewer = {John C. Wood},
      coden = {JDGEAS},
      title = {Flat {$G$}-bundles with canonical metrics},
      year = {1988},
      pages = {361--382},
      }
  • [CorletteArchimedean] Go to document K. Corlette, "Archimedean superrigidity and hyperbolic geometry," Ann. of Math., vol. 135, iss. 1, pp. 165-182, 1992.
    @ARTICLE{CorletteArchimedean, mrkey = {1147961},
      number = {1},
      issn = {0003-486X},
      author = {Corlette, Kevin},
      mrclass = {57S30 (22E40 53C25 57M50 58E20)},
      doi = {10.2307/2946567},
      journal = {Ann. of Math.},
      zblnumber = {0768.53025},
      volume = {135},
      mrnumber = {1147961},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Christopher W. Stark},
      coden = {ANMAAH},
      title = {Archimedean superrigidity and hyperbolic geometry},
      year = {1992},
      pages = {165--182},
      }
  • [Dani] Go to document S. G. Dani, "A simple proof of Borel’s density theorem," Math. Z., vol. 174, iss. 1, pp. 81-94, 1980.
    @ARTICLE{Dani, mrkey = {0591617},
      number = {1},
      issn = {0025-5874},
      author = {Dani, Shrikrishna G.},
      mrclass = {22D40 (20G20 58F17)},
      doi = {10.1007/BF01215084},
      journal = {Math. Z.},
      zblnumber = {0432.22008},
      volume = {174},
      mrnumber = {0591617},
      fjournal = {Mathematische Zeitschrift},
      mrreviewer = {Paul Milnes},
      coden = {MAZEAX},
      title = {A simple proof of {B}orel's density theorem},
      year = {1980},
      pages = {81--94},
      }
  • [Fischer] Go to document G. Fischer, Complex Analytic Geometry, New York: Springer-Verlag, 1976, vol. 538.
    @BOOK{Fischer, mrkey = {0430286},
      author = {Fischer, Gerd},
      mrclass = {32-01},
      series = {Lecture Notes in Math.},
      address = {New York},
      publisher = {Springer-Verlag},
      volume = {538},
      mrnumber = {0430286},
      mrreviewer = {Andrew Markoe},
      title = {Complex Analytic Geometry},
      year = {1976},
      pages = {vii+201},
      zblnumber = {0343.32002},
      doi = {10.1007/BFb0080338},
      }
  • [God] C. Godbillon, Feuilletages. Études Géométriques, Basel: Birkhäuser, 1991.
    @BOOK{God, mrkey = {1120547},
      author = {Godbillon, Claude},
      mrclass = {57R30 (58A17 58F18)},
      series = {Progr. Math.},
      isbn = {3-7643-2638-7},
      address = {Basel},
      publisher = {Birkhäuser},
      vollume = {98},
      mrnumber = {1120547},
      mrreviewer = {J. Chrastina},
      title = {Feuilletages. {É}tudes G{é}om{é}triques},
      year = {1991},
      pages = {xiv+474},
      zblnumber = {0724.58002},
      }
  • [GoldmanThesis] Go to document W. M. Goldman, Discontinuous Groups and the Euler Class, Ann Arbor, MI: ProQuest LLC, 1980.
    @BOOK{GoldmanThesis, mrkey = {2630832},
      author = {Goldman, William Mark},
      mrclass = {Thesis},
      url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:8029408},
      address = {Ann Arbor, MI},
      publisher = {ProQuest LLC},
      mrnumber = {2630832},
      note = {Thesis (Ph.D.)--University of California, Berkeley},
      title = {Discontinuous Groups and the {E}uler Class},
      year = {1980},
      pages = {138},
      }
  • [GoldmanComponents] Go to document W. M. Goldman, "Topological components of spaces of representations," Invent. Math., vol. 93, iss. 3, pp. 557-607, 1988.
    @ARTICLE{GoldmanComponents, mrkey = {0952283},
      number = {3},
      issn = {0020-9910},
      author = {Goldman, William M.},
      mrclass = {57M05 (22E40 32G15)},
      doi = {10.1007/BF01410200},
      journal = {Invent. Math.},
      zblnumber = {0655.57019},
      volume = {93},
      mrnumber = {0952283},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {William Harvey},
      coden = {INVMBH},
      title = {Topological components of spaces of representations},
      year = {1988},
      pages = {557--607},
      }
  • [GoldmanMillsonLocalRigidity] Go to document W. M. Goldman and J. J. Millson, "Local rigidity of discrete groups acting on complex hyperbolic space," Invent. Math., vol. 88, iss. 3, pp. 495-520, 1987.
    @ARTICLE{GoldmanMillsonLocalRigidity, mrkey = {0884798},
      number = {3},
      issn = {0020-9910},
      author = {Goldman, W. M. and Millson, J. J.},
      mrclass = {22E40 (57S25)},
      doi = {10.1007/BF01391829},
      journal = {Invent. Math.},
      zblnumber = {0627.22012},
      volume = {88},
      mrnumber = {0884798},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {G. A. So{\u\i}fer},
      coden = {INVMBH},
      title = {Local rigidity of discrete groups acting on complex hyperbolic space},
      year = {1987},
      pages = {495--520},
      }
  • [GriffithsSchmid] Go to document P. Griffiths and W. Schmid, "Locally homogeneous complex manifolds," Acta Math., vol. 123, pp. 253-302, 1969.
    @ARTICLE{GriffithsSchmid, mrkey = {0259958},
      issn = {0001-5962},
      author = {Griffiths, Phillip and Schmid, Wilfried},
      mrclass = {57.60 (32.00)},
      doi = {10.1007/BF02392390},
      journal = {Acta Math.},
      zblnumber = {0209.25701},
      volume = {123},
      mrnumber = {0259958},
      fjournal = {Acta Mathematica},
      mrreviewer = {S. Kobayashi},
      title = {Locally homogeneous complex manifolds},
      year = {1969},
      pages = {253--302},
      }
  • [Gromov1] Go to document M. Gromov, "Foliated Plateau problem. I: Minimal varieties," Geom. Funct. Anal., vol. 1, iss. 1, pp. 14-79, 1991.
    @ARTICLE{Gromov1, mrkey = {1091610},
      number = {1},
      issn = {1016-443X},
      author = {Gromov, M.},
      mrclass = {53C23 (53C12 58E15)},
      doi = {10.1007/BF01895417},
      journal = {Geom. Funct. Anal.},
      zblnumber = {0768.53011},
      volume = {1},
      mrnumber = {1091610},
      fjournal = {Geometric and Functional Analysis},
      mrreviewer = {Thomas H. Otway},
      coden = {GFANFB},
      title = {Foliated {P}lateau problem. {I}: {M}inimal varieties},
      year = {1991},
      pages = {14--79},
      }
  • [Gromov2] Go to document M. Gromov, "Foliated Plateau problem. II: Harmonic maps of foliations," Geom. Funct. Anal., vol. 1, iss. 3, pp. 253-320, 1991.
    @ARTICLE{Gromov2, mrkey = {1118731},
      number = {3},
      issn = {1016-443X},
      author = {Gromov, M.},
      mrclass = {58E20 (49Q99 53C12)},
      doi = {10.1007/BF01896204},
      journal = {Geom. Funct. Anal.},
      zblnumber = {0768.53012},
      volume = {1},
      mrnumber = {1118731},
      fjournal = {Geometric and Functional Analysis},
      mrreviewer = {Viktor Schroeder},
      coden = {GFANFB},
      title = {Foliated {P}lateau problem. {II}: {H}armonic maps of foliations},
      year = {1991},
      pages = {253--320},
      }
  • [GPS] Go to document M. Gromov and I. Piatetski-Shapiro, "Nonarithmetic groups in Lobachevsky spaces," Inst. Hautes Études Sci. Publ. Math., iss. 66, pp. 93-103, 1988.
    @ARTICLE{GPS, mrkey = {0932135},
      number = {66},
      issn = {0073-8301},
      author = {Gromov, M. and Piatetski-Shapiro, I.},
      mrclass = {22E40},
      url = {http://www.numdam.org/numdam-bin/item?id=PMIHES_1987__66__93_0},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      zblnumber = {0649.22007},
      mrnumber = {0932135},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      mrreviewer = {Gopal Prasad},
      title = {Nonarithmetic groups in {L}obachevsky spaces},
      year = {1988},
      pages = {93--103},
      }
  • [GromovSchoen] Go to document M. Gromov and R. Schoen, "Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one," Inst. Hautes Études Sci. Publ. Math., vol. 76, pp. 165-246, 1992.
    @ARTICLE{GromovSchoen, mrkey = {1215595},
      volume = {76},
      issn = {0073-8301},
      author = {Gromov, Mikhail and Schoen, Richard},
      mrclass = {58E20 (22E40)},
      url = {http://www.numdam.org/item?id=PMIHES_1992__76__165_0},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      zblnumber = {0896.58024},
      mrnumber = {1215595},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      mrreviewer = {Caio J. C. Negreiros},
      coden = {PMIHA6},
      title = {Harmonic maps into singular spaces and {$p$}-adic superrigidity for lattices in groups of rank one},
      year = {1992},
      pages = {165--246},
      }
  • [GW] Go to document O. Guichard and A. Wienhard, "Anosov representations: domains of discontinuity and applications," Invent. Math., vol. 190, iss. 2, pp. 357-438, 2012.
    @ARTICLE{GW, mrkey = {2981818},
      number = {2},
      issn = {0020-9910},
      author = {Guichard, Olivier and Wienhard, Anna},
      mrclass = {22F30 (32G15 53C30 53D25)},
      doi = {10.1007/s00222-012-0382-7},
      journal = {Invent. Math.},
      zblnumber = {1270.20049},
      volume = {190},
      mrnumber = {2981818},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {Pablo Su{á}rez-Serrato},
      coden = {INVMBH},
      title = {Anosov representations: domains of discontinuity and applications},
      year = {2012},
      pages = {357--438},
      }
  • [Hernandez] Go to document L. Hernández, "Maximal representations of surface groups in bounded symmetric domains," Trans. Amer. Math. Soc., vol. 324, iss. 1, pp. 405-420, 1991.
    @ARTICLE{Hernandez, mrkey = {1033234},
      number = {1},
      issn = {0002-9947},
      author = {Hern{á}ndez, Luis},
      mrclass = {32M15 (22E40)},
      doi = {10.2307/2001515},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {0733.32024},
      volume = {324},
      mrnumber = {1033234},
      fjournal = {Transactions of the American Mathematical Society},
      mrreviewer = {Jean-Jacques Loeb},
      coden = {TAMTAM},
      title = {Maximal representations of surface groups in bounded symmetric domains},
      year = {1991},
      pages = {405--420},
      }
  • [JohnsonMillson] Go to document D. Johnson and J. J. Millson, "Deformation spaces associated to compact hyperbolic manifolds," in Discrete Groups in Geometry and Analysis, Boston: Birkhäuser, 1987, vol. 67, pp. 48-106.
    @INCOLLECTION{JohnsonMillson, mrkey = {0900823},
      author = {Johnson, Dennis and Millson, John J.},
      mrclass = {22E40 (22E41 32G13 32M15)},
      series = {Progr. Math.},
      address = {Boston},
      publisher = {Birkhäuser},
      doi = {10.1007/978-1-4899-6664-3_3},
      zblnumber = {0664.53023},
      volume = {67},
      mrnumber = {0900823},
      booktitle = {Discrete Groups in Geometry and Analysis},
      mrreviewer = {T. N. Venkataramana},
      venue = {{N}ew {H}aven, {C}onn., 1984},
      title = {Deformation spaces associated to compact hyperbolic manifolds},
      pages = {48--106},
      year = {1987},
      }
  • [Klingler] Go to document B. Klingler, "Local rigidity for complex hyperbolic lattices and Hodge theory," Invent. Math., vol. 184, iss. 3, pp. 455-498, 2011.
    @ARTICLE{Klingler, mrkey = {2800692},
      number = {3},
      issn = {0020-9910},
      author = {Klingler, B.},
      mrclass = {22E40 (14C30)},
      doi = {10.1007/s00222-010-0293-4},
      journal = {Invent. Math.},
      zblnumber = {1239.22009},
      volume = {184},
      mrnumber = {2800692},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {William Goldman},
      coden = {INVMBH},
      title = {Local rigidity for complex hyperbolic lattices and {H}odge theory},
      year = {2011},
      pages = {455--498},
      }
  • [K] Go to document S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton, NJ: Princeton Univ. Press, 1987, vol. 15.
    @BOOK{K, mrkey = {0909698},
      author = {Kobayashi, Shoshichi},
      mrclass = {53C55 (32-02 32L05 32L10 32L20)},
      series = {Publ. Math. Soc. Japan},
      isbn = {0-691-08467-X},
      address = {Princeton, NJ},
      publisher = {Princeton Univ. Press},
      doi = {10.1515/9781400858682},
      zblnumber = {0708.53002},
      volume = {15},
      mrnumber = {0909698},
      note = {Kan{ô} Memorial Lectures, 5},
      mrreviewer = {Daniel M. Burns, Jr.},
      title = {Differential Geometry of Complex Vector Bundles},
      year = {1987},
      pages = {xii+305},
      }
  • [KozMobRank1] Go to document V. Koziarz and J. Maubon, "Harmonic maps and representations of non-uniform lattices of ${ PU}(m,1)$," Ann. Inst. Fourier $($Grenoble$)$, vol. 58, iss. 2, pp. 507-558, 2008.
    @ARTICLE{KozMobRank1, mrkey = {2410381},
      number = {2},
      issn = {0373-0956},
      author = {Koziarz, Vincent and Maubon, Julien},
      mrclass = {22E40 (53C24 58E20)},
      journal = {Ann. Inst. Fourier $($Grenoble$)$},
      zblnumber = {1147.22009},
      volume = {58},
      mrnumber = {2410381},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      mrreviewer = {John R. Parker},
      coden = {AIFUA7},
      title = {Harmonic maps and representations of non-uniform lattices of {${\rm PU}(m,1)$}},
      year = {2008},
      pages = {507--558},
      doi = {10.5802/aif.2359},
      }
  • [KozMobRank2] Go to document V. Koziarz and J. Maubon, "Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type," Geom. Dedicata, vol. 137, pp. 85-111, 2008.
    @ARTICLE{KozMobRank2, mrkey = {2449147},
      issn = {0046-5755},
      author = {Koziarz, Vincent and Maubon, Julien},
      mrclass = {22E40 (32L05 53C24)},
      doi = {10.1007/s10711-008-9288-3},
      journal = {Geom. Dedicata},
      zblnumber = {1159.22006},
      volume = {137},
      mrnumber = {2449147},
      fjournal = {Geometriae Dedicata},
      mrreviewer = {Anna Wienhard},
      coden = {GEMDAT},
      title = {Representations of complex hyperbolic lattices into rank 2 classical {L}ie groups of {H}ermitian type},
      year = {2008},
      pages = {85--111},
      }
  • [KozMobequidistrib] V. Koziarz and J. Maubon, On the equidistribution of totally geodesic submanifolds in locally symmetric spaces and application to boundedness results for negative curves and exceptional divisors, 2014.
    @MISC{KozMobequidistrib,
      author = {Koziarz, Vincent and Maubon, Julien},
      arxiv = {1407.6561},
      title = {On the equidistribution of totally geodesic submanifolds in locally symmetric spaces and application to boundedness results for negative curves and exceptional divisors},
      year = {2014},
      }
  • [Margulis] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, New York: Springer-Verlag, 1991, vol. 17.
    @BOOK{Margulis, mrkey = {1090825},
      author = {Margulis, G. A.},
      mrclass = {22E40 (20Hxx 22-02 22D40)},
      series = {Ergeb. Math. Grenzgeb.},
      address = {New York},
      isbn = {3-540-12179-X},
      publisher = {Springer-Verlag},
      volume = {17},
      mrnumber = {1090825},
      mrreviewer = {Gopal Prasad},
      title = {Discrete Subgroups of Semisimple {L}ie Groups},
      year = {1991},
      pages = {x+388},
      zblnumber = {0732.22008},
      }
  • [MarkmanXia] Go to document E. Markman and E. Z. Xia, "The moduli of flat ${ PU}(p,p)$-structures with large Toledo invariants," Math. Z., vol. 240, iss. 1, pp. 95-109, 2002.
    @ARTICLE{MarkmanXia, mrkey = {1906709},
      number = {1},
      issn = {0025-5874},
      author = {Markman, E. and Xia, E. Z.},
      mrclass = {14D20 (14H60)},
      doi = {10.1007/s002090100364},
      journal = {Math. Z.},
      zblnumber = {1008.32006},
      volume = {240},
      mrnumber = {1906709},
      fjournal = {Mathematische Zeitschrift},
      mrreviewer = {Steven B. Bradlow},
      coden = {MAZEAX},
      title = {The moduli of flat {${\rm PU}(p,p)$}-structures with large {T}oledo invariants},
      year = {2002},
      pages = {95--109},
      }
  • [Maubon] J. Maubon, "Higgs bundles and representations of complex hyperbolic lattices," in Handbook of Group Actions. Vol. II, Int. Press, Somerville, MA, 2015, vol. 32, pp. 201-244.
    @INCOLLECTION{Maubon, mrkey = {3382029},
      author = {Maubon, Julien},
      mrclass = {32G20 (32L05 32M15 53C07 53C43)},
      series = {Adv. Lect. Math. (ALM)},
      publisher = {Int. Press, Somerville, MA},
      volume = {32},
      mrnumber = {3382029},
      booktitle = {Handbook of Group Actions. {V}ol. {II}},
      mrreviewer = {Daniel Greb},
      title = {Higgs bundles and representations of complex hyperbolic lattices},
      pages = {201--244},
      year = {2015},
      }
  • [Mok] Go to document N. Mok, "On holomorphic immersions into Kähler manifolds of constant holomorphic sectional curvature," Sci. China Ser. A, vol. 48, iss. suppl., pp. 123-145, 2005.
    @ARTICLE{Mok, mrkey = {2156495},
      number = {suppl.},
      issn = {1006-9283},
      author = {Mok, Ngaiming},
      mrclass = {32Q15 (32Q30 53C55)},
      doi = {10.1007/BF02884700},
      journal = {Sci. China Ser. A},
      zblnumber = {1128.32014},
      volume = {48},
      mrnumber = {2156495},
      fjournal = {Science in China. Series A. Mathematics},
      mrreviewer = {Qi Lin Yang},
      title = {On holomorphic immersions into {K}ähler manifolds of constant holomorphic sectional curvature},
      year = {2005},
      pages = {123--145},
      }
  • [P] Go to document T. L. Payne, "Closures of totally geodesic immersions into locally symmetric spaces of noncompact type," Proc. Amer. Math. Soc., vol. 127, iss. 3, pp. 829-833, 1999.
    @ARTICLE{P, mrkey = {1468202},
      number = {3},
      issn = {0002-9939},
      author = {Payne, Tracy L.},
      mrclass = {53C30 (53C35 53C42)},
      doi = {10.1090/S0002-9939-99-04552-9},
      journal = {Proc. Amer. Math. Soc.},
      zblnumber = {0936.53034},
      volume = {127},
      mrnumber = {1468202},
      fjournal = {Proceedings of the American Mathematical Society},
      mrreviewer = {Raul Quiroga},
      coden = {PAMYAR},
      title = {Closures of totally geodesic immersions into locally symmetric spaces of noncompact type},
      year = {1999},
      pages = {829--833},
      }
  • [Per] Go to document J. V. Pereira, "Global stability for holomorphic foliations on Kaehler manifolds," Qual. Theory Dyn. Syst., vol. 2, iss. 2, pp. 381-384, 2001.
    @ARTICLE{Per, mrkey = {1913291},
      number = {2},
      issn = {1575-5460},
      author = {Pereira, J. V.},
      mrclass = {32S65 (32Q15 37F75)},
      doi = {10.1007/BF02969347},
      journal = {Qual. Theory Dyn. Syst.},
      zblnumber = {1074.53019},
      volume = {2},
      mrnumber = {1913291},
      fjournal = {Qualitative Theory of Dynamical Systems},
      mrreviewer = {M. G. Soares},
      title = {Global stability for holomorphic foliations on {K}aehler manifolds},
      year = {2001},
      pages = {381--384},
      }
  • [Pop] Go to document D. Popovici, "A simple proof of a theorem by Uhlenbeck and Yau," Math. Z., vol. 250, iss. 4, pp. 855-872, 2005.
    @ARTICLE{Pop, mrkey = {2180378},
      number = {4},
      issn = {0025-5874},
      author = {Popovici, Dan},
      mrclass = {32L05 (32C30 32L10 32Q15)},
      doi = {10.1007/s00209-005-0780-2},
      journal = {Math. Z.},
      zblnumber = {1083.32018},
      volume = {250},
      mrnumber = {2180378},
      fjournal = {Mathematische Zeitschrift},
      mrreviewer = {Julien Keller},
      coden = {MAZEAX},
      title = {A simple proof of a theorem by {U}hlenbeck and {Y}au},
      year = {2005},
      pages = {855--872},
      }
  • [Pozzetti] Go to document M. B. Pozzetti, "Maximal representations of complex hyperbolic lattices into ${ SU}(M,N)$," Geom. Funct. Anal., vol. 25, iss. 4, pp. 1290-1332, 2015.
    @ARTICLE{Pozzetti, mrkey = {3385634},
      number = {4},
      issn = {1016-443X},
      author = {Pozzetti, Maria Beatrice},
      mrclass = {32M05 (22E40 32M15 32Q45)},
      doi = {10.1007/s00039-015-0338-3},
      journal = {Geom. Funct. Anal.},
      zblnumber = {1325.22007},
      volume = {25},
      mrnumber = {3385634},
      fjournal = {Geometric and Functional Analysis},
      title = {Maximal representations of complex hyperbolic lattices into {${\rm SU}(M,N)$}},
      year = {2015},
      pages = {1290--1332},
      }
  • [Raghunathan] M. S. Raghunathan, Discrete Subgroups of Lie Groups, New York: Springer-Verlag, 1972, vol. 68.
    @BOOK{Raghunathan, mrkey = {0507234},
      author = {Raghunathan, M. S.},
      mrclass = {22E40},
      series = {Ergeb. Math. Grenzgeb.},
      address = {New York},
      publisher = {Springer-Verlag},
      volume = {68},
      mrnumber = {0507234},
      mrreviewer = {J. S. Joel},
      title = {Discrete Subgroups of {L}ie Groups},
      year = {1972},
      pages = {ix+227},
      zblnumber = {0254.22005},
      }
  • [RatnerMeasure] Go to document M. Ratner, "On Raghunathan’s measure conjecture," Ann. of Math., vol. 134, iss. 3, pp. 545-607, 1991.
    @ARTICLE{RatnerMeasure, mrkey = {1135878},
      number = {3},
      issn = {0003-486X},
      author = {Ratner, Marina},
      mrclass = {22E40 (58F11 58F17)},
      doi = {10.2307/2944357},
      journal = {Ann. of Math.},
      zblnumber = {0763.28012},
      volume = {134},
      mrnumber = {1135878},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {S. G. Dani},
      coden = {ANMAAH},
      title = {On {R}aghunathan's measure conjecture},
      year = {1991},
      pages = {545--607},
      }
  • [RatnerTopo] Go to document M. Ratner, "Raghunathan’s topological conjecture and distributions of unipotent flows," Duke Math. J., vol. 63, iss. 1, pp. 235-280, 1991.
    @ARTICLE{RatnerTopo, mrkey = {1106945},
      number = {1},
      issn = {0012-7094},
      author = {Ratner, Marina},
      mrclass = {22E40 (22D40 28D10)},
      doi = {10.1215/S0012-7094-91-06311-8},
      journal = {Duke Math. J.},
      zblnumber = {0733.22007},
      volume = {63},
      mrnumber = {1106945},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {Gopal Prasad},
      coden = {DUMJAO},
      title = {Raghunathan's topological conjecture and distributions of unipotent flows},
      year = {1991},
      pages = {235--280},
      }
  • [Richardson] Go to document R. W. Richardson, "Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups," Duke Math. J., vol. 57, iss. 1, pp. 1-35, 1988.
    @ARTICLE{Richardson, mrkey = {0952224},
      number = {1},
      issn = {0012-7094},
      author = {Richardson, R. W.},
      mrclass = {20G15 (14L30 17B45 22E46)},
      doi = {10.1215/S0012-7094-88-05701-8},
      journal = {Duke Math. J.},
      zblnumber = {0685.20035},
      volume = {57},
      mrnumber = {0952224},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {Michel Brion},
      coden = {DUMJAO},
      title = {Conjugacy classes of {$n$}-tuples in {L}ie algebras and algebraic groups},
      year = {1988},
      pages = {1--35},
      }
  • [Ro80] Go to document H. L. Royden, "The Ahlfors-Schwarz lemma in several complex variables," Comment. Math. Helv., vol. 55, iss. 4, pp. 547-558, 1980.
    @ARTICLE{Ro80, mrkey = {0604712},
      number = {4},
      issn = {0010-2571},
      author = {Royden, H. L.},
      mrclass = {32H20},
      doi = {10.1007/BF02566705},
      journal = {Comment. Math. Helv.},
      zblnumber = {0484.53053},
      volume = {55},
      mrnumber = {0604712},
      fjournal = {Commentarii Mathematici Helvetici},
      mrreviewer = {Marcus Wright},
      coden = {COMHAX},
      title = {The {A}hlfors-{S}chwarz lemma in several complex variables},
      year = {1980},
      pages = {547--558},
      }
  • [Sampson] Go to document J. H. Sampson, "Applications of harmonic maps to Kähler geometry," in Complex Differential Geometry and Nonlinear Differential Equations, Providence, RI: Amer. Math. Soc., 1986, vol. 49, pp. 125-134.
    @INCOLLECTION{Sampson, mrkey = {0833809},
      author = {Sampson, J. H.},
      mrclass = {58E20 (32C10 53C55)},
      series = {Contemp. Math.},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      doi = {10.1090/conm/049/833809},
      zblnumber = {0605.58019},
      volume = {49},
      mrnumber = {0833809},
      booktitle = {Complex Differential Geometry and Nonlinear Differential Equations},
      mrreviewer = {Toru Ishihara},
      venue = {{B}runswick, {M}aine, 1984},
      title = {Applications of harmonic maps to {K}ähler geometry},
      pages = {125--134},
      year = {1986},
      }
  • [Shah] Go to document N. A. Shah, "Uniformly distributed orbits of certain flows on homogeneous spaces," Math. Ann., vol. 289, iss. 2, pp. 315-334, 1991.
    @ARTICLE{Shah, mrkey = {1092178},
      number = {2},
      issn = {0025-5831},
      author = {Shah, Nimish A.},
      mrclass = {22E40 (58F11)},
      doi = {10.1007/BF01446574},
      journal = {Math. Ann.},
      zblnumber = {0702.22014},
      volume = {289},
      mrnumber = {1092178},
      fjournal = {Mathematische Annalen},
      mrreviewer = {T. N. Venkataramana},
      coden = {MAANA},
      title = {Uniformly distributed orbits of certain flows on homogeneous spaces},
      year = {1991},
      pages = {315--334},
      }
  • [Sibley] Go to document B. Sibley, "Asymptotics of the Yang-Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit," J. Reine Angew. Math., vol. 706, pp. 123-191, 2015.
    @ARTICLE{Sibley, mrkey = {3393366},
      issn = {0075-4102},
      author = {Sibley, Benjamin},
      mrclass = {53C44 (53C55 58E15)},
      doi = {10.1515/crelle-2013-0063},
      journal = {J. Reine Angew. Math.},
      zblnumber = {1329.58006},
      volume = {706},
      mrnumber = {3393366},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      mrreviewer = {Yu Zheng},
      title = {Asymptotics of the {Y}ang-{M}ills flow for holomorphic vector bundles over {K}ähler manifolds: the canonical structure of the limit},
      year = {2015},
      pages = {123--191},
      }
  • [S1] Go to document C. T. Simpson, "Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization," J. Amer. Math. Soc., vol. 1, iss. 4, pp. 867-918, 1988.
    @ARTICLE{S1, mrkey = {0944577},
      number = {4},
      issn = {0894-0347},
      author = {Simpson, Carlos T.},
      mrclass = {58E15 (32L15 53C25 53C55)},
      doi = {10.2307/1990994},
      journal = {J. Amer. Math. Soc.},
      zblnumber = {0669.58008},
      volume = {1},
      mrnumber = {0944577},
      fjournal = {Journal of the American Mathematical Society},
      title = {Constructing variations of {H}odge structure using {Y}ang-{M}ills theory and applications to uniformization},
      year = {1988},
      pages = {867--918},
      }
  • [S2] Go to document C. T. Simpson, "Higgs bundles and local systems," Inst. Hautes Études Sci. Publ. Math., vol. 75, pp. 5-95, 1992.
    @ARTICLE{S2, mrkey = {1179076},
      volume = {75},
      issn = {0073-8301},
      author = {Simpson, Carlos T.},
      mrclass = {32G13 (14D07 53C07 58D27 58E15)},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      zblnumber = {0814.32003},
      mrnumber = {1179076},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      mrreviewer = {William Goldman},
      coden = {PMIHA6},
      title = {Higgs bundles and local systems},
      year = {1992},
      pages = {5--95},
      doi = {10.1007/BF02699491},
      }
  • [Siu] Go to document Y. T. Siu, "The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds," Ann. of Math., vol. 112, iss. 1, pp. 73-111, 1980.
    @ARTICLE{Siu, mrkey = {0584075},
      number = {1},
      issn = {0003-486X},
      author = {Siu, Yum Tong},
      mrclass = {53C55 (32H99 58E20)},
      doi = {10.2307/1971321},
      journal = {Ann. of Math.},
      zblnumber = {0517.53058},
      volume = {112},
      mrnumber = {0584075},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {M. Kalka},
      coden = {ANMAAH},
      title = {The complex-analyticity of harmonic maps and the strong rigidity of compact {K}ähler manifolds},
      year = {1980},
      pages = {73--111},
      }
  • [Sullivan] Go to document D. Sullivan, "Cycles for the dynamical study of foliated manifolds and complex manifolds," Invent. Math., vol. 36, pp. 225-255, 1976.
    @ARTICLE{Sullivan, mrkey = {0433464},
      issn = {0020-9910},
      author = {Sullivan, Dennis},
      mrclass = {57D15},
      doi = {10.1007/BF01390011},
      journal = {Invent. Math.},
      zblnumber = {0335.57015},
      volume = {36},
      mrnumber = {0433464},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {Robert Roussarie},
      title = {Cycles for the dynamical study of foliated manifolds and complex manifolds},
      year = {1976},
      pages = {225--255},
      }
  • [ToledoHarmonic] Go to document D. Toledo, "Harmonic maps from surfaces to certain Kaehler manifolds," Math. Scand., vol. 45, iss. 1, pp. 13-26, 1979.
    @ARTICLE{ToledoHarmonic, mrkey = {0567429},
      number = {1},
      issn = {0025-5521},
      author = {Toledo, Domingo},
      mrclass = {58E20 (32H30 53C55)},
      journal = {Math. Scand.},
      zblnumber = {0435.58008},
      volume = {45},
      mrnumber = {0567429},
      fjournal = {Mathematica Scandinavica},
      mrreviewer = {Samuel I. Goldberg},
      coden = {MTSCAN},
      title = {Harmonic maps from surfaces to certain {K}aehler manifolds},
      year = {1979},
      pages = {13--26},
      url = {http://www.mscand.dk/article/view/11821/9837},
     }
  • [ToledoRepresentations] Go to document D. Toledo, "Representations of surface groups in complex hyperbolic space," J. Differential Geom., vol. 29, iss. 1, pp. 125-133, 1989.
    @ARTICLE{ToledoRepresentations, mrkey = {0978081},
      number = {1},
      issn = {0022-040X},
      author = {Toledo, Domingo},
      mrclass = {57N05 (22E40 32G13 57S25 58E20)},
      url = {http://projecteuclid.org/euclid.jdg/1214442638},
      journal = {J. Differential Geom.},
      zblnumber = {0676.57012},
      volume = {29},
      mrnumber = {0978081},
      fjournal = {Journal of Differential Geometry},
      mrreviewer = {Robert Brooks},
      coden = {JDGEAS},
      title = {Representations of surface groups in complex hyperbolic space},
      year = {1989},
      pages = {125--133},
      }
  • [UhlenbeckYau] Go to document K. Uhlenbeck and S. -T. Yau, "On the existence of Hermitian-Yang-Mills connections in stable vector bundles," Comm. Pure Appl. Math., vol. 39, p. s257-s293, 1986.
    @ARTICLE{UhlenbeckYau, mrkey = {0861491},
      issn = {0010-3640},
      author = {Uhlenbeck, K. and Yau, S.-T.},
      mrclass = {58G05 (32L15 53C05 58E15)},
      doi = {10.1002/cpa.3160390714},
      journal = {Comm. Pure Appl. Math.},
      zblnumber = {0615.58045},
      volume = {39},
      mrnumber = {0861491},
      fjournal = {Communications on Pure and Applied Mathematics},
      mrreviewer = {Daniel S. Freed},
      coden = {CPAMA},
      title = {On the existence of {H}ermitian-{Y}ang-{M}ills connections in stable vector bundles},
      year = {1986},
      pages = {S257--S293},
      }
  • [westwick] Go to document R. Westwick, "Spaces of linear transformations of equal rank," Linear Algebra and Appl., vol. 5, pp. 49-64, 1972.
    @ARTICLE{westwick, mrkey = {0296081},
      author = {Westwick, R.},
      mrclass = {15A03},
      journal = {Linear Algebra and Appl.},
      zblnumber = {0236.15002},
      volume = {5},
      mrnumber = {0296081},
      mrreviewer = {J. E. Whitesitt},
      title = {Spaces of linear transformations of equal rank},
      year = {1972},
      pages = {49--64},
      doi = {10.1016/0024-3795(72)90018-3},
      }
  • [Wolf] Go to document J. A. Wolf, "The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components," Bull. Amer. Math. Soc., vol. 75, pp. 1121-1237, 1969.
    @ARTICLE{Wolf, mrkey = {0251246},
      issn = {0002-9904},
      author = {Wolf, Joseph A.},
      mrclass = {32.32 (22.00)},
      doi = {10.1090/S0002-9904-1969-12359-1},
      journal = {Bull. Amer. Math. Soc.},
      zblnumber = {0183.50901},
      volume = {75},
      mrnumber = {0251246},
      fjournal = {Bulletin of the American Mathematical Society},
      mrreviewer = {W. Klingenberg},
      title = {The action of a real semisimple group on a complex flag manifold. {I}. {O}rbit structure and holomorphic arc components},
      year = {1969},
      pages = {1121--1237},
      }
  • [Xia] Go to document E. Z. Xia, "The moduli of flat ${ PU}(2,1)$ structures on Riemann surfaces," Pacific J. Math., vol. 195, iss. 1, pp. 231-256, 2000.
    @ARTICLE{Xia, mrkey = {1781622},
      number = {1},
      issn = {0030-8730},
      author = {Xia, Eugene Z.},
      mrclass = {32G13 (14H60 53C07)},
      doi = {10.2140/pjm.2000.195.231},
      journal = {Pacific J. Math.},
      zblnumber = {1014.32010},
      volume = {195},
      mrnumber = {1781622},
      fjournal = {Pacific Journal of Mathematics},
      mrreviewer = {Usha N. Bhosle},
      coden = {PJMAAI},
      title = {The moduli of flat {${\rm PU}(2,1)$} structures on {R}iemann surfaces},
      year = {2000},
      pages = {231--256},
      }
  • [You] Go to document D. C. Youla, "A normal form for a matrix under the unitary congruence group," Canad. J. Math., vol. 13, pp. 694-704, 1961.
    @ARTICLE{You, mrkey = {0132754},
      issn = {0008-414X},
      author = {Youla, D. C.},
      mrclass = {15.30},
      doi = {10.4153/CJM-1961-059-8},
      journal = {Canad. J. Math.},
      zblnumber = {0103.25201},
      volume = {13},
      mrnumber = {0132754},
      fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      mrreviewer = {O. Taussky-Todd},
      title = {A normal form for a matrix under the unitary congruence group},
      year = {1961},
      pages = {694--704},
      }

Authors

Vincent Koziarz

Univ. Bordeaux, IMB, CNRS, UMR 5251, F-33400 Talence, France

Julien Maubon

Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France