American Mathematical Society

Book Review

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MathSciNet review: 1567883
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Book Information:

Authors: Ravi S. Kulkarni and Ulrich Pinkall
Title: Conformal geometry
Additional book information: Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunsweig and Wiesbaden, 1988, viii + 236 pp. ISBN 3-528-08982-2.

B. N. Apanasov, Nontriviality of Teichmüller space for Kleinian group in space, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 21–31. MR 624802

[A2] B. N. Apanosov, On non-connectedness of the space of uniformized conformal structures on hyperbolic manifolds, Institut Mittag-Leffler, Djursholm Report, no. 1, 1989-1990.

Robert Brooks, The fundamental group and the spectrum of the Laplacian, Comment. Math. Helv. 56 (1981), no. 4, 581–598. MR 656213, DOI 10.1007/BF02566228

Jean-Pierre Conze and Yves Guivarc’h, Remarques sur la distalité dans les espaces vectoriels, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1083–1086 (French). MR 339108

D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852

David Fried, Closed similarity manifolds, Comment. Math. Helv. 55 (1980), no. 4, 576–582. MR 604714, DOI 10.1007/BF02566707

[FL] F. Luo, Triangulations in conformally flat geometry, USCD, preprint.

William M. Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for $3$-manifolds, Trans. Amer. Math. Soc. 278 (1983), no. 2, 573–583. MR 701512, DOI 10.1090/S0002-9947-1983-0701512-8

M. Gromov, H. B. Lawson Jr., and W. Thurston, Hyperbolic $4$-manifolds and conformally flat $3$-manifolds, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 27–45 (1989). MR 1001446

Dennis Johnson and John J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 48–106. MR 900823, DOI 10.1007/978-1-4899-6664-3_{3}

Yoshinobu Kamishima, Conformally flat manifolds whose development maps are not surjective. I, Trans. Amer. Math. Soc. 294 (1986), no. 2, 607–623. MR 825725, DOI 10.1090/S0002-9947-1986-0825725-2

M. È. Kapovich, Flat conformal structures on three-dimensional manifolds: the existence problem. I, Sibirsk. Mat. Zh. 30 (1989), no. 5, 60–73, 216 (Russian); English transl., Siberian Math. J. 30 (1989), no. 5, 712–722 (1990). MR 1025290, DOI 10.1007/BF00971261

[KP] M. Kapovich and L. Potyagailo, The absence of Ahlfors'finiteness theorem in higher dimensions, Novosibirsk preprint, 1989.

Christos Kourouniotis, Deformations of hyperbolic structures, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 2, 247–261. MR 795891, DOI 10.1017/S030500410006343X

N. H. Kuiper, On conformally-flat spaces in the large, Ann. of Math. (2) 50 (1949), 916–924. MR 31310, DOI 10.2307/1969587

N. H. Kuiper, On compact conformally Euclidean spaces of dimension $>2$, Ann. of Math. (2) 52 (1950), 478–490. MR 37575, DOI 10.2307/1969480

Nicolaas H. Kuiper, Hyperbolic $4$-manifolds and tesselations, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 47–76 (1989). MR 1001447

R. S. Kulkarni, On the principle of uniformization, J. Differential Geometry 13 (1978), no. 1, 109–138. MR 520605

[L] F. Labourie, Surfaces convexes dans l'espace hyperbolique et CP (submitted).

Shigenori Matsumoto, Foundations of flat conformal structure, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math., vol. 20, Kinokuniya, Tokyo, 1992, pp. 167–261. MR 1208312, DOI 10.2969/aspm/02010167

Madhav V. Nori, The Schottky groups in higher dimensions, The Lefschetz centennial conference, Part I (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1986, pp. 195–197. MR 860414, DOI 10.1090/conm/058.1/860414

R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71. MR 931204, DOI 10.1007/BF01393992

Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401

Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833

Dennis Sullivan and William Thurston, Manifolds with canonical coordinate charts: some examples, Enseign. Math. (2) 29 (1983), no. 1-2, 15–25. MR 702731

[T] W. Thurston, The geometry and topology of 3-manifolds, Princeton Univ. lecture notes, 1979.

Review Information:

Reviewer: William M. Goldman

Journal: Bull. Amer. Math. Soc. 23 (1990), 566-575
DOI: https://doi.org/10.1090/S0273-0979-1990-15984-1