A pinching theorem for cusps of negatively curved manifolds with finite volume

Kanai, Masahiko

doi:10.1090/S0002-9939-1989-0937856-5

A pinching theorem for cusps of negatively curved manifolds with finite volume
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Proc. Amer. Math. Soc. 107 (1989), 777-783 Request permission

Abstract:

We give a new proof of the following theorem of M. Gromov: For a noncompact complete riemannian manifold $M$ of negative curvature with finite volume, each cusp of $M$ is diffeomorphic to $N \times [0,\infty )$ with $N$ being a compact flat space form provided that the sectional curvature of $M$ satisfies the pinching condition $- 4 < - {\Lambda ^2} \leq K \leq - 1$.

References

Thierry Aubin, Espaces de Sobolev sur les variétés riemanniennes, Bull. Sci. Math. (2) 100 (1976), no. 2, 149–173 (French). MR 488125
Peter Buser and Hermann Karcher, Gromov’s almost flat manifolds, Astérisque, vol. 81, Société Mathématique de France, Paris, 1981. MR 619537
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
Patrick Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), no. 3, 435–476. MR 577132, DOI 10.2307/1971104
Leon W. Green, The generalized geodesic flow, Duke Math. J. 41 (1974), 115–126; correction, ibid. 42 (1975), 381. MR 370659
M. Gromov, Manifolds of negative curvature, J. Differential Geometry 13 (1978), no. 2, 223–230. MR 540941
M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978), no. 2, 231–241. MR 540942
M. Gromov, Synthetic geometry in Riemannian manifolds, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 415–419. MR 562635
M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12. MR 892185, DOI 10.1007/BF01404671
Ernst Heintze and Hans-Christoph Im Hof, Geometry of horospheres, J. Differential Geometry 12 (1977), no. 4, 481–491 (1978). MR 512919
Masahiko Kanai, Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations, Ergodic Theory Dynam. Systems 8 (1988), no. 2, 215–239. MR 951270, DOI 10.1017/S0143385700004430
Masahiko Kanai, Tensorial ergodicity of geodesic flows, Geometry and analysis on manifolds (Katata/Kyoto, 1987) Lecture Notes in Math., vol. 1339, Springer, Berlin, 1988, pp. 142–157. MR 961479, DOI 10.1007/BFb0083053
Pierre Pansu, Quasiconformal mappings and manifolds of negative curvature, Curvature and topology of Riemannian manifolds (Katata, 1985) Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 212–229. MR 859587, DOI 10.1007/BFb0075658
Takashi Sakai, Comparison and finiteness theorems in Riemannian geometry, Geometry of geodesics and related topics (Tokyo, 1982) Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1984, pp. 125–181. MR 758652, DOI 10.2969/aspm/00310125
Viktor Schroeder, Finite volume and fundamental group on manifolds of negative curvature, J. Differential Geom. 20 (1984), no. 1, 175–183. MR 772130
Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. MR 1001277

Complete negatively curved Kähler surfaces of finite volume

Marina Ville, Sur les variétés riemanniennes ${1\over 4}$-pincées de dimension $4$ et de courbure négative, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 12, 397–400 (French, with English summary). MR 794747