Journal of Differential Geometry

Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties

Jürgen Jost and Kang Zuo

Full-text: Open access

Article information

Source
J. Differential Geom., Volume 47, Number 3 (1997), 469-503.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1214460547

Digital Object Identifier
doi:10.4310/jdg/1214460547

Mathematical Reviews number (MathSciNet)
MR1617644

Zentralblatt MATH identifier
0911.58012

Subjects
Primary: 58E20: Harmonic maps [See also 53C43], etc.
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

Citation

Jost, Jürgen; Zuo, Kang. Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties. J. Differential Geom. 47 (1997), no. 3, 469--503. doi:10.4310/jdg/1214460547. https://projecteuclid.org/euclid.jdg/1214460547


Export citation

References

  • [1] S.I. Al'ber, On n-dimensional problems in the calculus of variations in the large, Soviet. Math. Dokl. 5 (1964) 700-704.
  • [2] S.I. Al'ber, Spaces of mappings into a manifold with negative curvature, Soviet. Math. Dokl. 9 (1967) 6-9.
  • [3] A. Borel and J.P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436-491.
  • [4] K. Corlette, Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988) 361-382.
  • [5] K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. 135 (1992) 165-182.
  • [6] M. Cornalba and P. Gri ths, Analytic cycles and vector bundles on non compact algebraic varieties, Invent. Math. 28 (1975) 1-106.
  • [7] J. Carlson and D. Toledo, Harmonic mappings of Kahler manifolds to locally symmetric spaces, Inst. Hautes Etudes Sci. Publ. Math. 69 (1989) 173-201.
  • [8] S. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987) 127-131.
  • [9] K. Diederich and T. Ohsawa, Harmonic mappings and disk bundles over compact Kahler manifolds, Publ. Res. Inst. Math. Sci. 21 (1985) 819-833.
  • [10] J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 85 (1964) 109-160.
  • [11] M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Inst. Hautes Etudes Sci. Publ. Math. 76 (1992) 165-246.
  • [12] P. Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967) 673-687.
  • [13] N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59-126.
  • [14] S. Iitaka, Logarithmic forms on algebraic varieties, J. Fac. Sci., Univ. Tokyo, Sect. 1A Math., 23 (1976) 525-544.
  • [15] J. Jost, Equilibrium maps between metric spaces, Calculus Variations 2 (1994) 173-204.
  • [16] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comm. Math. Helv. 70 (1995) 659-673.
  • [17] J. Jost, Riemannian geometry and geometric analysis, Springer, Berlin, 1995.
  • [18] J. Jost, Generalized harmonic maps between metric spaces, (ed. J. Jost), Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt, Internat. Press, 1996, 143-174.
  • [19] J. Jost and S. T. Yau, Harmonic mappings and Kahler manifolds, Math. Ann. 262 (1983) 145-166.
  • [20] J. Jost and S. T. Yau, The strong rigidity of locally symmetric complex manifolds of rank one and nite volume, Math. Ann. 275 (1986) 291-304.
  • [21] J. Jost and S. T. Yau, On the rigidity of certain discrete groups and algebraic varieties, Math. Ann. 278 (1987) 481-496.
  • [22] J. Jost and S. T. Yau, Harmonic maps and group representations, Differential Geometry and Minimal Submanifolds, (B. Lawson and K. Tenenblat eds.), Longman Scientific, 1991, 241-260.
  • [23] J. Jost and S. T. Yau, Harmonic maps and superrigidity, Proc. Sympos. Pure Math. 54 (1993) Part I, 245-280.
  • [24] J. Jost and K. Zuo, Harmonic maps and Sl(r C)-representations of fundamental groups of quasi projective manifolds, J. Algebraic Geom. 5 (1996) 77-106.
  • [25] J. Jost and K. Zuo, Harmonic maps into Tits buildings and factorization of non rigid and non arithmetic representations of 1 of algebraic varieties, Preprint, 1993.
  • [26] J. Jost and K. Zuo, Harmonic maps of in nite energy and rigidity results for quasiprojective varieties, Math. Res. Lett. 1 (1994) 631-638.
  • [27] L. Katzarkov, Factorization theorems for the representations of the fundamental groups of quasiprojective varieties and some applications, Preprint.
  • [28] N. Korevaar and R. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993) 561-659.
  • [29] J. Kollar, Shavarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993) 165-215.
  • [30] F. Labourie, Existence d'applicationsharmoniquestordues a valeurs dans les varietes a courbure negative, Proc. Amer. Math. Soc. 111 (1991) 877-882.
  • [31] J. Y. Li, Hitchin's self-duality equations on complete Riemannian manifolds, Math. Ann., to appear.
  • [32] J. Lohkamp, An existence theorem for harmonic maps, Manuscripta Math. 67 (1990) 21-23.
  • [33] G. A. Margulis, Discrete groups of motion of manifolds of nonpositive curvature, Trans. Amer. Math. Soc. 190 (1977) 33-45.
  • [34] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Springer, Berlin, 1991.
  • [35] J. Morgan, ;-trees and their applications, Bull. Amer. Math. Soc. 26 (1992) 87-112.
  • [36] G. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Stud. 78, Princeton, 1973.
  • [37] N. Mok, Y.-T. Siu and S.-K. Yeung, Geometric superrigidity, Invent. Math. 113 (1993) 57-84.
  • [38] G. Prasad, Strong rigidity of Q-rank 1 lattices, Invent. Math. 21 (1973) 255-286.
  • [39] H. Royden, The Ahlfors-Schwarz lemma in several complex variables, Comment. Math. Helv. 55 (1980) 547-558.
  • [40] C. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990) 713-769.
  • [41] C. Simpson, A Lefschetz theorem for 0 of the integral leaves of a holomorphic oneform, Compositio Math. 87 (1993) 99-113.
  • [42] C. Simpson, Higgs bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992) 5-95.
  • [43] C. Simpson, Moduli of representation of the fundamental group of a smooth projective variety I and II, Inst. Hautes Etudes Sci. Publ. Math.
  • [44] J. Sampson, Applications of harmonic maps to Kahler geometry, Contemp. Math. 49 (1986) 125-134.
  • [45] R. Schoen, Thesis, Stanford, 1978.
  • [46] Y. T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds, Ann. of Math. 112 (1980) 73-111.
  • [47] Y. T. Siu, Strong rigidity of compact quotients of exceptional symmetric domains, Duke Math. J. 48 (1981) 857-871.
  • [48] M Wolf, In nite energy maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991) 487-539.
  • [49] S. T. Yau, A general Schwarz lemma for Kahler manifolds, Amer. J. Math. 100 (1978) 197-203.
  • [50] K. Zuo, Some structure theorems for semi-simple representations of 1 of algebraic manifolds, Math. Ann. 295 (1993) 365-382.
  • [51] K. Zuo, Factorization of nonrigid Zariski dense representation of 1 of projective algebraic manifolds, Invent. Math. 118 (1994) 37-46. Max-Planck-Institute for Mathematics in the Sciences University Kaiserslautern, Germany