Variational problems for riemannian functionals and arithmetic groups
Publications Mathématiques de l'IHÉS, Volume 92 (2000), pp. 5-62.
@article{PMIHES_2000__92__5_0,
     author = {Nabutovsky, Alexander and Weinberger, Shmuel},
     title = {Variational problems for riemannian functionals and arithmetic groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {5--62},
     publisher = {Institut des Hautes \'Etudes Scientifiques},
     volume = {92},
     year = {2000},
     zbl = {1003.58007},
     mrnumber = {1839486},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2000__92__5_0/}
}
TY  - JOUR
AU  - Nabutovsky, Alexander
AU  - Weinberger, Shmuel
TI  - Variational problems for riemannian functionals and arithmetic groups
JO  - Publications Mathématiques de l'IHÉS
PY  - 2000
DA  - 2000///
SP  - 5
EP  - 62
VL  - 92
PB  - Institut des Hautes Études Scientifiques
UR  - http://www.numdam.org/item/PMIHES_2000__92__5_0/
UR  - https://zbmath.org/?q=an%3A1003.58007
UR  - https://www.ams.org/mathscinet-getitem?mr=1839486
LA  - en
ID  - PMIHES_2000__92__5_0
ER  - 
%0 Journal Article
%A Nabutovsky, Alexander
%A Weinberger, Shmuel
%T Variational problems for riemannian functionals and arithmetic groups
%J Publications Mathématiques de l'IHÉS
%D 2000
%P 5-62
%V 92
%I Institut des Hautes Études Scientifiques
%G en
%F PMIHES_2000__92__5_0
Nabutovsky, Alexander; Weinberger, Shmuel. Variational problems for riemannian functionals and arithmetic groups. Publications Mathématiques de l'IHÉS, Volume 92 (2000), pp. 5-62. http://www.numdam.org/item/PMIHES_2000__92__5_0/

[AVS] D. Alekseevskij, E. Vinberg, A. Solodovnikov, Geometry of spaces of constant curvature, in Geometry II, ed. by E. Vinberg, Encyclopaedia of Math. Sci., vol. 29, Springer, 1993. | MR | Zbl

[An 1] M. T. Anderson, Degeneration of metrics with bounded curvature and applications to critical metrics of Riemannian functionals, in Differential Geometry: Riemannian Geometry, Proceedings of symposia in pure mathematics, ed. by R. Greene and S. T. Yau, 54:3 (1993), 53-79. | MR | Zbl

[An 2] M. T. Anderson, Scalar curvature and geometrization conjectures for 3-manifolds, in Comparison Geometry, ed. by K. Grove and P. Petersen, 49-82, MSRI Publications, 1997. | MR | Zbl

[BT] E. Ballico, A. Tognoli, Algebraic models defined over Q of differentiable manifolds, Geom. Dedicata, 42 (1992), 155-162. | MR | Zbl

[BGS] W. Ballman, M. Gromov, V. Scroeder, Manifolds of non-positive sectional curvature, Birkhauser, Boston, 1985.

[Ba] S. Bando, Real Analyticity of solutions of Hamilton's equation, Math. Z., 195 (1987), 93-97. | MR | Zbl

[B] J. M. Barzdin, Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set, Soviet Math. Dokl. 9 (1968), 1251-1254. | Zbl

[BC] P. Baum, A. Connes, Chern character for discrete groups, in A fête of topology, 163-232, Academic Press, 1988. | MR | Zbl

[BCH] P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C*-algebras, in C*-algebras: 1943-1993 (San Antonio, TX, 1993), 240-291, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994. | MR | Zbl

[Baum] G. Baumslag, Topics in combinatorial group theory, Boston, Birkhauser, 1993. | MR | Zbl

[BMR] J. Bemelmans, M. Min-Oo, E. Ruh, Smoothing Riemannian metrics, Math. Z., 188 (1984), 69-74. | MR | Zbl

[BP] R. Benedetti, C. Petronio, Lectures on hyperbolic geometry, Springer, 1992. | MR | Zbl

[BN] V. N. Berestovskij, I.G. Nikolaev, Multidimensional generalized Riemannian spaces, in Geometry IV, ed. Yu. G. Reshetnyak, Encyclopaedia of Mathematical Sciences, vol. 70, Springer, 1993. | MR | Zbl

[Br] M. Berger, Recent trends in Riemannian geometry, in Recent trends in mathematics, Reinhardsbrunn 1982, 20-37, Leipzig, Teubner, 1982. | Zbl

[Be] A. Besse, Einstein manifolds, Springer, 1987. | MR | Zbl

[BW] J. Block, S. Weinberger, Arithmetic manifolds with positive scalar curvature, preprint. | Zbl

[BCR] J. Bochnak, M. Coste, M.-F. Roy, Geometrie Algebrique Reelle, Springer, 1987. | MR | Zbl

[BHP] W. Boone, W. Haken, V. Poenaru, On recursively unsolvable problems in topology and their classification, in Contributions to mathematical logic (H. Arnold Schmidt, K. Schutte, H.-J. Thiele, eds.), North-Holland, 1968. | MR | Zbl

[Bor 1] A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111-122. | MR | Zbl

[Bor 2] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sc. École Norm. Sup. 7 (1974), 235-272. | Numdam | MR | Zbl

[Bor 3] A. Borel, Stable real cohomology of arithmetic groups II, in Manifolds and Lie groups, ed. by J. Hano et al., 21-56, Boston, Birkhauser, 1981. | MR | Zbl

[BorWal] A. Borel, N. Wallach, Continuous cohomology, discrete subgroups and representations of reductive groups, Ann. of Math. Studies 94, Princeton Univ. Press, 1980. | MR | Zbl

[BY] A. Borel, J. Yang, The rank conjecture for number fields, Math. Res. Lett. 1 (1994), 689-699. | MR | Zbl

[Bou] J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Comp. Math., 30 (1975), 1-41. | Numdam | MR | Zbl

[BGP] Yu. Burago, M. Gromov, G. Perelman, A. D. Alexandrov spaces with curvature bounded from below, Russ. Math. Surv. 47 (1992), 1-58. | Zbl

[CS] S. Cappell, J. Shaneson, The codimension two placement problem and the homology equivalent manifolds, Ann. Math. 99 (1974), 277-348. | MR | Zbl

[Ch] I. Chavel, Riemannian geometry: a modern introduction, Cambridge University Press, 1995. | Zbl

[C] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61-74. | MR | Zbl

[CMS] J. Cheeger, W. Muller, R. Schrader, On the curvature of piecewise-flat spaces, Comm. Math. Phys. 92 (1984), 405-454. | MR | Zbl

[CGT] J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom. 17 (1982), 15-53. | MR | Zbl

[C10] L. Clozel, Représentations galoisiennes associées aux représentations automorphes autoduales de GL(n), Publications IHES, 73 (1991), 97-145. | Numdam | MR | Zbl

[C1] L. Clozel, On the cohomology of Kottwitz's arithmetic varieties, Duke Math. J., 72 (1993), 757-795. | MR | Zbl

[CoSh] M. Coste, M. Shiota, Nash triviality for families of Nash manifolds, Inv. Math. 108 (1992), 349-368. | MR | Zbl

[D] R. D. Daley, Minimal-program complexity of sequences with restricted resources, Inf. and Control 23 (1973), 301-312. | MR | Zbl

[FHT] Y. Felix, S. Halperin, J.-C. Thomas, Rational Homotopy Theory, to appear. | Zbl

[F] K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, in Recent Topics in Differential and Analytic Geometry, ed. T. Ochiai, Adv. Stud. Pure Math. 18-I, Boston, Academic Press, 1990. | MR | Zbl

[GM] P. Griffiths, J. W. Morgan, Rational Homotopy Theory and Differential Forms, Birkhauser, 1981. | MR | Zbl

[Gr 1] M. Gromov, Volume and bounded cohomology, Publications IHES 56 (1982), 5-99. | Numdam | MR | Zbl

[Gr 2] M. Gromov, Filling of Riemannian manifolds, J. Differential Geom., 18 (1983), 1-148. | MR | Zbl

[GL 1] M. Gromov, H. B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. Math. 111 (1980), 209-230. | MR | Zbl

[GL 2] M. Gromov, H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publications IHES, 58 (1983), 295-408. | Numdam | MR | Zbl

[GP] K. Grove, P. Petersen V, Bounding homotopy type by geometry, Ann. Math. 128 (1988), 195-206. | MR | Zbl

[Ha] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geometry 17 (1982), 255-306. | MR | Zbl

[Haus] J.-P. Hausmann, Manifolds with a given homology and fundamental group, Commun. Math. Helv. 53 (1978), 113-134. | MR | Zbl

[H] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, New York, Academic Press, 1978. | Zbl

[I] N. V. Ivanov, Foundations of the theory of bounded cohomology, J. of Soviet Math., 37 (1987), 1090-1115. | Zbl

[JK] J. Jost, H. Karcher, Geometrische Methoden zur Gewinnung von A-Priori-Schranken fur harmonische Abbildungen, Manuscripta Math., 40 (1982), 27-78. | MR | Zbl

[KS] A. Karrass, D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227-255. | MR | Zbl

[KaS] G. Kasparov, G. Skandalis, Groups acting on buildings, operator K-theory, and Novikov's conjecture, K-theory, 4 (1991), 303-337. | MR | Zbl

[K0] M. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67-72. | MR | Zbl

[K] M. Kervaire, Multiplicateurs de Schur et K-theories, in Essays in topology and related topics, ed. A. Haefliger and R. Narashimhan, Springer, 1970, 212-225. | MR | Zbl

[Kt0] R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), 611-650. | MR | Zbl

[Kt1] R. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365-399. | MR | Zbl

[L] J. Lohkamp, Global and local curvatures, in Riemannian Geometry, ed. by M. Lovric, M. Min-Oo, M. Wang, Fields Institute Monograph, 4, American Mathematical Society, 1996, 23-52. | MR | Zbl

[LS] R. C. Lyndon, P. E. Schupp, Combinatorial group theory, Springer, 1977. | MR | Zbl

[LV] M. Li, P. M. B. Vitanyi, Kolmogorov complexity and its applications, in Handbook of theoretical computer science, ed. by Jan van Leeuwen, Elsevier, 1990, 187-254. | MR | Zbl

[Ma] G. Margulis, Arithmetic groups, Springer, 1991.

[M] C. Miller, Decision problems for groups-survey and reflections, in Combinatorial group theory (G. Baumslag, C. Miller, eds.), Springer 1989, 1-59. | MR | Zbl

[Mil 1] J. Milnor, Introduction to algebraic K-theory, Annals of Mathematical Studies, Princeton Univ. Press, 1971. | MR | Zbl

[Mil 2] J. Milnor, On the homology of Lie groups made discrete, Comm. Math. Helv., 58 (1983), 72-85. | MR | Zbl

[N1] A. Nabutovsky, Einstein structures: existence versus uniqueness, Geom. Funct. Anal. 5 (1995), 76-91. | MR | Zbl

[N2] A. Nabutovsky, Geometry of the space of triangulations of a compact manifold, Commun. Math. Phys., 181 (1996), 303-330. | MR | Zbl

[N3] A. Nabutovsky, Disconnectedness of sublevel sets of some Riemannian functionals, Geom. Funct. Anal. 6 (1996), 703-725. | MR | Zbl

[N4] A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. on Pure and Appl. Math. 49 (12) (1996), 1257-1270. | MR | Zbl

[P] P. Petersen V, Gromov-Hausdorff convergence of metric spaces, in Differential Geometry: Riemannian Geometry, ed. by R. Greene, S. T. Yau, Proceedings of AMS Symposia in Pure Mathematics, 54:3 (1993), 489-504. | MR | Zbl

[Pi1] M. Pimsner, KK-groups of crossed products by groups acting on trees, Invent. Math., 86 (1986), 603-634. | MR | Zbl

[Pi2] M. Pimsner, K-theory for groups acting on trees, Proceedings of the Int. Congr. of Math., Kyoto, 1990, 979-986, Math. Soc. Japan, 1991. | Zbl

[R] J. J. Rotman, An introduction to the theory of groups, Springer, 1995. | MR | Zbl

[Ros 1] J. Rosenberg, C*-algebras, positive scalar curvature, and the Novikov conjecture, Publications IHES, 58 (1983), 197-212. | Numdam | MR | Zbl

[Ros 2] J. Rosenberg, Algebraic K-theory and its applications, Springer, 1993.

[Sr] P. Sarnak, Extremal geometries, in Extremal Riemann surfaces, ed. by J. R. Quine and Peter Sarnak, Contemp. Math. 201, 1-7, AMS, Providence, RI, 1997. | MR | Zbl

[Ser] J.-P. Serre, Arithmetic groups, in Homological group theory, ed. by C. T. C. Wall, 105-136, Cambridge, Cambridge Univ. Press, 1979. | MR | Zbl

[Sch] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Springer, LNM 1365 (1989), 120-154. | MR | Zbl

[S] D. Sullivan, Infinitesimal computations in topology, Publications IHES 47 (1977), 269-331. | Numdam | MR | Zbl

[T] A. Thompson, Thin position and the recognition problem for S3, Math. Res. Lett. 1 (1994), 613-630. | MR | Zbl

[V] P. Vogel, Un théorème d'Hurewicz homologique, Commun. Math. Helv. 52 (1977), 393-413. | MR | Zbl

[ZL] A. K. Zvonkin, L. A. Levin, The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms, Russ. Math. Surv. 25 (6) (1970), 83-129. | MR | Zbl

[Y] S. T. Yau, Open problems in geometry, in Differential Geometry: Riemannian Geometry, ed. by R. Greene and S. T. Yau, Proceedings of AMS Symposia in Pure Mathematics, 54:1 (1993), 1-28. | MR | Zbl