Harmonic maps into singular spaces and p -adic superrigidity for lattices in groups of rank one

Michael Gromov; Richard Schoen

Publications Mathématiques de l'IHÉS (1992)

  • Volume: 76, page 165-246
  • ISSN: 0073-8301

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Gromov, Michael, and Schoen, Richard. "Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one." Publications Mathématiques de l'IHÉS 76 (1992): 165-246. <http://eudml.org/doc/104083>.

@article{Gromov1992,
author = {Gromov, Michael, Schoen, Richard},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {-adic representations; -adic superrigidity; harmonic mappings; nonpositively curved metric spaces; lattices in noncompact semisimple Lie groups; arithmeticity for lattices; isometry groups; quaternionic hyperbolic space; Cayley plane},
language = {eng},
pages = {165-246},
publisher = {Institut des Hautes Études Scientifiques},
title = {Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one},
url = {http://eudml.org/doc/104083},
volume = {76},
year = {1992},
}

TY - JOUR
AU - Gromov, Michael
AU - Schoen, Richard
TI - Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one
JO - Publications Mathématiques de l'IHÉS
PY - 1992
PB - Institut des Hautes Études Scientifiques
VL - 76
SP - 165
EP - 246
LA - eng
KW - -adic representations; -adic superrigidity; harmonic mappings; nonpositively curved metric spaces; lattices in noncompact semisimple Lie groups; arithmeticity for lattices; isometry groups; quaternionic hyperbolic space; Cayley plane
UR - http://eudml.org/doc/104083
ER -

References

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  1. [ABB] S. B. ALEXANDER, I. D. BERG and R. L. BISHOP, The Riemannian obstacle problem, III. J. Math. 31 (1987), 167-184. Zbl0625.53045MR88a:53038
  2. [Ag] S. AGMON, Unicité et convexité dans les problèmes différentiels, Sém. d'Analyse Sup., Univ. de Montréal, 1965. Zbl0147.07702
  3. [Al] F. J. ALMGREN, Jr., Q-valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two, preprint, Princeton University. 
  4. [B] K. BROWN, Buildings, Springer, New York, 1989. Zbl0715.20017MR90e:20001
  5. [BT] F. BRUHAT and J. TITS, Groupes réductifs sur un corps local. I. Données radicielles valuées, Publ. Math. IHES 41 (1972), 5-251. Zbl0254.14017MR48 #6265
  6. [C] K. CORLETTE, Archimedian superrigidity and hyperbolic geometry, Annals of Math., to appear. Zbl0768.53025
  7. [Ch] Y. J. CHIANG, Harmonic maps of V-manifolds, Ann. Global Anal. Geom. 8 (1990), 315-344. Zbl0679.58014MR92c:58021
  8. [DM] P. DELIGNE and G. D. MOSTOW, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. IHES 63 (1986), 5-89. Zbl0615.22008MR88a:22023a
  9. [ES] J. EELLS and J. H. SAMPSON, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. Zbl0122.40102MR29 #1603
  10. [F1] H. FEDERER, Geometric Measure Theory, Springer-Verlag, New York, 1969. Zbl0176.00801MR41 #1976
  11. [F2] H. FEDERER, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 79 (1970), 761-771. Zbl0194.35803MR41 #5601
  12. [G] H. GARLAND, P-adic curvature and the cohomology of discrete subgroups, Ann. of Math. 97 (1973), 375-423. Zbl0262.22010MR47 #8719
  13. [GL] N. GAROFALO and F. H. LIN, Monotonicity properties of variational integrals, Ap weights and unique continuation, Indiana Math. J. 35 (1986), 245-268. Zbl0678.35015MR88b:35059
  14. [GP] M. GROMOV, P, PANSU, Rigidity of lattices: An introduction, to appear in Springer Lecture Notes. Zbl0786.22015
  15. [GPS] M. GROMOV and I. PIATETSKI-SHAPIRO, Non-arithmetic groups in Lobachevsky spaces, Publ. Math. IHES 66 (1988), 93-103. Zbl0649.22007MR89j:22019
  16. [GR] H. GARLAND and M. S. RAGHUNATHAN, Fundamental domains for lattices in R-rank 1 groups, Ann. of Math. 92 (1970), 279-326. Zbl0206.03603MR42 #1943
  17. [Gro] M. GROMOV, Partial differential relations, Springer Verlag, 1986. Zbl0651.53001MR90a:58201
  18. [Ham] R. HAMILTON, Harmonic maps of manifolds with boundary, Lecture Notes 471, Springer 1975. Zbl0308.35003MR58 #2872
  19. [Har] P. HARTMAN, On homotopic harmonic maps, Can. J. Math. 19 (1967), 673-687. Zbl0148.42404MR35 #4856
  20. [HV] P. de la HARPE and A. VALETTE, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175 (1989), Soc. Math. de France. Zbl0759.22001
  21. [K] H. KARCHER, Riemannian center of mass and mollifier smoothing, Comm. Pure and Appl. Math. 30 (1977), 509-541. Zbl0354.57005MR56 #1350
  22. [La1] E. M. LANDIS, A three-sphere theorem, Dokl. Akad. Nauk S.S.S.R. 148 (1963), 277-279. Translated in Soviet Math. 4 (1963), 76-78. Zbl0145.14302MR27 #443
  23. [La2] E. M. LANDIS, Some problems on the qualitative theory of second order elliptic equations (case of several variables), Uspekhi Mat. Nauk. 18 (1963), 3-62. Translated in Russian Math. Surveys 18 (1963), 1-62. Zbl0125.05802
  24. [L] M. L. LEITE, Harmonic mappings of surfaces with respect to degenerate metrics, Amer. J. Math. 110 (1988), 399-412. Zbl0646.58018MR89h:58046
  25. [Lin] F. H. LIN, Nonlinear theory of defects in nematic liquid crystals, phase transition and flow phenomena, Comm. Pure Appl. Math. 42 (1989), 789-814. Zbl0703.35173MR90g:82076
  26. [Mak] V. MAKAROV, On a certain class of discrete Lobachevsky space groups with infinite fundamental domain of finite measure, Soviet Math. Dokl. 7 (1966), 328-331. Zbl0146.16502MR36 #3566
  27. [Mar] G. MARGULIS, Discrete groups of motions of manifolds of nonpositive curvature, AMS Translations 109 (1977), 33-45. Zbl0367.57012
  28. [Mos] G. D. MOSTOW, On a remarkable class of polyhedra in complex hyperbolic space, Pac. J. Math. 86 (1980), 171-276. Zbl0456.22012MR82a:22011
  29. [Mi] K. MILLER, Three circles theorems in partial differential equations and applications to improperly posed problems, Arch. for Rat. Mech. and Anal. 16 (1964), 126-154. Zbl0145.14203MR29 #1435
  30. [Mo] C. B. MORREY, Multiple integrals in the calculus of variations, Springer-Verlag, New York, 1966. Zbl0142.38701MR34 #2380
  31. [N] I. G. NIKOLAEV, Solution of Plateau problem in spaces with curvature ≤ K, Sib, Math. J. 20 : 2 (1979), 346-353. Zbl0434.53045MR80k:58041
  32. [S] R. SCHOEN, Analytic aspects of the harmonic map problem, Math. Sci. Res. Inst. Publ. vol. 2, Springer, Berlin, 1984, 321-358. Zbl0551.58011MR86b:58032
  33. [Se] A. SELBERG, Recent developments in the theory of discontinuous groups of motions of symmetric spaces, Springer Lecture Notes 118 (1970), 99-120. Zbl0197.18002MR41 #8595
  34. [Ser] J. P. SERRE, Trees, Springer Verlag, 1980. Zbl0548.20018MR82c:20083
  35. [Sim] C. SIMPSON, Integrality of rigid local systems of rank two on a smooth projective variety, preprint. 
  36. [Siu] Y. T. SIU, The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980), 73-112. Zbl0517.53058MR81j:53061
  37. [Ste] K. STEIN, Analytische Zerlegungen komplexer Räume, Math. Ann. 132 (1956), 63-93. Zbl0074.06301MR18,649c
  38. [SU] R. SCHOEN and K. UHLENBECK, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307-335. Zbl0521.58021MR84b:58037a
  39. [SY] R. SCHOEN and S. T. YAU, Harmonic maps and the topology of stable hyper-surfaces and manifolds of nonnegative Ricci curvature, Comment. Math. Helv. 39 (1976), 333-341. Zbl0361.53040MR55 #11302
  40. [V] E. VINBERG, Discrete groups generated by reflections in Lobachevsky spaces, Math. USSR-Sb 1 (1967), 429-444. Zbl0166.16303
  41. [Z] W. P. ZIEMER, Weakly Differentiable Functions, Springer-Verlag, Grad. Texts in Math., 1989. Zbl0692.46022MR91e:46046
  42. [Zim] R. J. ZIMMER, Ergodic Theory and Semi-simple Groups, Birkhäuser, Boston, Basel, Stuttgart, 1984. Zbl0571.58015MR86j:22014

Citations in EuDML Documents

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  1. A. Majumdar, J. M. Robbins, M. Zyskin, Energies of S 2 -valued harmonic maps on polyhedra with tangent boundary conditions
  2. Édouard Lebeau, Applications harmoniques entre graphes finis et un théorème de superrigidité
  3. Édouard Lebeau, Applications harmoniques entre graphes finis et un théorème de superrigidité
  4. Thibaut Delcroix, Les groupes de Burger-Mozes ne sont pas kählériens
  5. Slavyana Geninska, On arithmetic Fuchsian groups and their characterizations
  6. Zahra Sinaei, Riemannian Polyhedra and Liouville-Type Theorems for Harmonic Maps
  7. Ngaiming Mok, Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups
  8. Brendon Lasell, Mohan Ramachandran, Observations on harmonic maps and singular varieties
  9. Katsutoshi Yamanoi, On fundamental groups of algebraic varieties and value distribution theory
  10. Terrence Napier, Mohan Ramachandran, [unknown]

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