Tubes in complex hyperbolic manifolds

Basmajian, Ara; Kim, Youngju

doi:10.1090/tran/9319

Tubes in complex hyperbolic manifolds
HTML articles powered by AMS MathViewer

by Ara Basmajian and Youngju Kim;

Trans. Amer. Math. Soc. 378 (2025), 2031-2060

DOI: https://doi.org/10.1090/tran/9319

Published electronically: November 6, 2024

HTML | PDF | Request permission

Abstract:

We prove a tubular neighborhood theorem for an embedded complex geodesic in a complex hyperbolic 2-manifold where the width of the tube depends only on the Euler characteristic $\chi$ of the embedded complex geodesic. We give an explicit estimate for this width. We supply two applications of the tubular neighborhood theorem. The first is a lower volume bound for such manifolds. The second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold. Finally, we prove a geometric combination theorem for two $\mathbb {C}$-Fuchsian subgroups of $\operatorname {PU}(2,1)$. Using this combination theorem, we show that the optimal width size of a tube about an embedded complex geodesic is asymptotically bounded between $\frac {1}{|\chi |}$ and $\frac {1}{\sqrt {|\chi |}}$.

References

Uri Bader, David Fisher, Nicholas Miller, and Matthew Stover, Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds, Invent. Math. 233 (2023), no. 1, 169–222. MR 4601998, DOI 10.1007/s00222-023-01186-5
Ara Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 1, 65–74. MR 1050782, DOI 10.5186/aasfm.1990.1505
Ara Basmajian, Generalizing the hyperbolic collar lemma, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 154–158. MR 1145576, DOI 10.1090/S0273-0979-1992-00298-7
Ara Basmajian, The stable neighborhood theorem and lengths of closed geodesics, Proc. Amer. Math. Soc. 119 (1993), no. 1, 217–224. MR 1152271, DOI 10.1090/S0002-9939-1993-1152271-0
Ara Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994), no. 2, 207–225. MR 1273264, DOI 10.1007/BF01232240
Ara Basmajian and Robert Miner, Discrete subgroups of complex hyperbolic motions, Invent. Math. 131 (1998), no. 1, 85–136. MR 1489895, DOI 10.1007/s002220050198
A. Basmajian, J. Brisson, A. Hassannezhad, and A. Metrás, Tubes and Steklov eigenvalues in negatively curved manifolds, arXiv.
Ara Basmajian and Scott A. Wolpert, Hyperbolic 3-manifolds with nonintersecting closed geodesics, Geom. Dedicata 97 (2003), 251–257. Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999). MR 2003700, DOI 10.1023/A:1023657806258
Jonas Beyrer and Beatrice Pozzetti, A collar lemma for partially hyperconvex surface group representations, Trans. Amer. Math. Soc. 374 (2021), no. 10, 6927–6961. MR 4315593, DOI 10.1090/tran/8453
Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
Robert Brooks and J. Peter Matelski, Collars in Kleinian groups, Duke Math. J. 49 (1982), no. 1, 163–182. MR 650375
Marc Burger and Viktor Schroeder, Volume, diameter and the first eigenvalue of locally symmetric spaces of rank one, J. Differential Geom. 26 (1987), no. 2, 273–284. MR 906391
Peter Buser, The collar theorem and examples, Manuscripta Math. 25 (1978), no. 4, 349–357. MR 509590, DOI 10.1007/BF01168048
Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
Wensheng Cao and John R. Parker, Jørgensen’s inequality and collars in $n$-dimensional quaternionic hyperbolic space, Q. J. Math. 62 (2011), no. 3, 523–543. MR 2825470, DOI 10.1093/qmath/haq003
Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Isaac Chavel and Edgar A. Feldman, Cylinders on surfaces, Comment. Math. Helv. 53 (1978), no. 3, 439–447. MR 493868, DOI 10.1007/BF02566089
Ted Chinburg and Matthew Stover, Geodesic curves on Shimura surfaces, Topology Proc. 52 (2018), 113–121. MR 3705536
Emily B. Dryden and Hugo Parlier, Collars and partitions of hyperbolic cone-surfaces, Geom. Dedicata 127 (2007), 139–149. MR 2338522, DOI 10.1007/s10711-007-9172-6
Daniel Gallo, A $3$-dimensional hyperbolic collar lemma, Kleinian groups and related topics (Oaxtepec, 1981) Lecture Notes in Math., vol. 971, Springer, Berlin-New York, 1983, pp. 31–35. MR 690276
F. W. Gehring and G. J. Martin, Inequalities for Möbius transformations and discrete groups, J. Reine Angew. Math. 418 (1991), 31–76. MR 1111201
Jane Gilman, A geometric approach to Jørgensen’s inequality, Adv. Math. 85 (1991), no. 2, 193–197. MR 1093005, DOI 10.1016/0001-8708(91)90055-C
William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR 1695450, DOI 10.1093/oso/9780198537939.001.0001
William M. Goldman, Michael Kapovich, and Bernhard Leeb, Complex hyperbolic manifolds homotopy equivalent to a Riemann surface, Comm. Anal. Geom. 9 (2001), no. 1, 61–95. MR 1807952, DOI 10.4310/CAG.2001.v9.n1.a3
Noemi Halpern, A proof of the collar lemma, Bull. London Math. Soc. 13 (1981), no. 2, 141–144. MR 608099, DOI 10.1112/blms/13.2.141
Sa’ar Hersonsky and Frédéric Paulin, On the volumes of complex hyperbolic manifolds, Duke Math. J. 84 (1996), no. 3, 719–737. MR 1408542, DOI 10.1215/S0012-7094-96-08422-7
F. Hirzebruch, Characteristic numbers of homogeneous domains, Seminars on Analytic Functions, Vol. Il, Institute for Advanced Studies, Princeton, 1957, pp. 92–104.
Linda Keen, Collars on Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Stud., No. 79, Princeton Univ. Press, Princeton, NJ, 1974, pp. 263–268. MR 379833
Daeyong Kim, Collar lemma in quaternionic hyperbolic manifold, Bull. Korean Math. Soc. 43 (2006), no. 2, 411–418. MR 2229537, DOI 10.4134/BKMS.2006.43.2.411
Sadayoshi Kojima, Immersed geodesic surfaces in hyperbolic $3$-manifolds, Complex Variables Theory Appl. 29 (1996), no. 1, 45–58. MR 1382002, DOI 10.1080/17476939608814873
Gye-Seon Lee and Tengren Zhang, Collar lemma for Hitchin representations, Geom. Topol. 21 (2017), no. 4, 2243–2280. MR 3654108, DOI 10.2140/gt.2017.21.2243
Elon L. Lima, The Jordan-Brouwer separation theorem for smooth hypersurfaces, Amer. Math. Monthly 95 (1988), no. 1, 39–42. MR 935429, DOI 10.2307/2323445
Sarah Markham and John R. Parker, Collars in complex and quaternionic hyperbolic manifolds, Geom. Dedicata 97 (2003), 199–213. Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999). MR 2003698, DOI 10.1023/A:1023677417179
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
Robert Meyerhoff, A lower bound for the volume of hyperbolic $3$-manifolds, Canad. J. Math. 39 (1987), no. 5, 1038–1056. MR 918586, DOI 10.4153/CJM-1987-053-6
J. R. Parker, Notes on Complex Hyperbolic Geometry.
John R. Parker, On the volumes of cusped, complex hyperbolic manifolds and orbifolds, Duke Math. J. 94 (1998), no. 3, 433–464. MR 1639519, DOI 10.1215/S0012-7094-98-09418-2
Jouni Parkkonen and Frédéric Paulin, A classification of $\Bbb C$-Fuchsian subgroups of Picard modular groups, Math. Scand. 121 (2017), no. 1, 57–74. MR 3708964, DOI 10.7146/math.scand.a-26128
Hugo Parlier, A note on collars of simple closed geodesics, Geom. Dedicata 112 (2005), 165–168. MR 2163896, DOI 10.1007/s10711-004-4661-3
John G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR 2249478
Richard Schoen, A lower bound for the first eigenvalue of a negatively curved manifold, J. Differential Geometry 17 (1982), no. 2, 233–238. MR 664495
Matthew Stover, Volumes of Picard modular surfaces, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3045–3056. MR 2811261, DOI 10.1090/S0002-9939-2011-10786-X
M. Stover, Residual finiteness and discrete subgroups of Lie groups, arXiv.

AMS Website Down

Transactions of the American Mathematical Society

Tubes in complex hyperbolic manifolds
HTML articles powered by AMS MathViewer

Abstract: