Geometry of Riemann surfaces based on closed geodesics

Schaller, Paul

doi:10.1090/S0273-0979-98-00750-2

Geometry of Riemann surfaces based on closed geodesics
HTML articles powered by AMS MathViewer

by Paul Schmutz Schaller PDF

Bull. Amer. Math. Soc. 35 (1998), 193-214 Request permission

Abstract:

The paper presents a survey on recent results on the geometry of Riemann surfaces showing that the study of closed geodesics provides a link between different aspects of Riemann surface theory such as hyperbolic geometry, topology, spectral theory, and the theory of arithmetic Fuchsian groups. Of particular interest are the systoles, the shortest closed geodesics of a surface; their study leads to the hyperbolic geometry of numbers with close analogues to classical sphere packing problems.

References

William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR 590044
C. Adams. Maximal cusps, collars and systoles in hyperbolic surfaces. To appear in Indiana Univ. Math. J.
Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Avner Ash, On eutactic forms, Canadian J. Math. 29 (1977), no. 5, 1040–1054. MR 491523, DOI 10.4153/CJM-1977-101-2
Avner Ash, On the existence of eutactic forms, Bull. London Math. Soc. 12 (1980), no. 3, 192–196. MR 572099, DOI 10.1112/blms/12.3.192
Christophe Bavard, Systole et invariant d’Hermite, J. Reine Angew. Math. 482 (1997), 93–120 (French). MR 1427658, DOI 10.1515/crll.1997.482.93
Anne-Marie Bergé and Jacques Martinet, Sur la classification des réseaux eutactiques, J. London Math. Soc. (2) 53 (1996), no. 3, 417–432 (French). MR 1396707, DOI 10.1112/jlms/53.3.417
Joan S. Birman and Caroline Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), no. 2, 217–225. MR 793185, DOI 10.1016/0040-9383(85)90056-4
Béla Bollobás, Extremal graph theory, London Mathematical Society Monographs, vol. 11, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 506522
Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, DOI 10.1007/BF01393996
Francis Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), no. 2, 233–297 (English, with English and French summaries). MR 1413855, DOI 10.5802/afst.829
A. Borel, Commensurability classes and volumes of hyperbolic $3$-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1–33. MR 616899
B. H. Bowditch, A proof of McShane’s identity via Markoff triples, Bull. London Math. Soc. 28 (1996), no. 1, 73–78. MR 1356829, DOI 10.1112/blms/28.1.73
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR 1324339
Peter Buser, Riemannsche Flächen mit Eigenwerten in $(0,$ $1/4)$, Comment. Math. Helv. 52 (1977), no. 1, 25–34. MR 434961, DOI 10.1007/BF02567355
Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3–92. MR 903850
J. W. S. Cassels, An introduction to the geometry of numbers, Die Grundlehren der mathematischen Wissenschaften, Band 99, Springer-Verlag, Berlin-New York, 1971. Second printing, corrected. MR 0306130
Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685, DOI 10.1017/CBO9780511623912
J. H. H. Chalk, Algebraic lattices, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 97–110. MR 731107
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
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
J.H. Conway; N.J.A.Sloane. Lattices with few distances. J. Number theory 39 (1991), 75-90.
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1993. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1194619, DOI 10.1007/978-1-4757-2249-9
Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, Mathematical Surveys and Monographs, vol. 30, American Mathematical Society, Providence, RI, 1989. MR 1010419, DOI 10.1090/surv/030
Max Dehn, Papers on group theory and topology, Springer-Verlag, New York, 1987. Translated from the German and with introductions and an appendix by John Stillwell; With an appendix by Otto Schreier. MR 881797, DOI 10.1007/978-1-4612-4668-8
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
Jozef Dodziuk, Thea Pignataro, Burton Randol, and Dennis Sullivan, Estimating small eigenvalues of Riemann surfaces, The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985) Contemp. Math., vol. 64, Amer. Math. Soc., Providence, RI, 1987, pp. 93–121. MR 881458, DOI 10.1090/conm/064/881458
P. Erdös, P. M. Gruber, and J. Hammer, Lattice points, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 39, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1003606
H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
A. Fathi; F. Laudenbach; V. Poénaru. Travaux de Thurston sur les surfaces. Séminaire Orsay, Astérisque 66-67 (1979).
L. Fejes Tóth. Kreisausfüllungen der hyperbolischen Ebene. Acta Math. Acad. Sci. Hungar 4 (1953), 103-110.
W. Fenchel; J. Nielsen. Discontinuous groups of non-euclidean motions. Unpublished manuscript (1948).
R. Fricke; F. Klein. Vorlesungen über die Theorie der elliptischen Modulfunktionen. Band 1,2. Teubner Leipzig (1890,1892).
F. Fricker. Lehrbuch der Gitterpunktlehre. Birkhäuser (1982).
D. Ginzburg; Z. Rudnick. Stable multiplicities in the length spectrum of Riemann surfaces. To appear in Israel J. Math.
D.D. Gorskov. Geometry of Lobachevski in connection with certain questions in arithmetic. J. Soviet Math. 16 (1981), 780-820.
P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. MR 893813
Laurent Guillopé, Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique, Duke Math. J. 53 (1986), no. 3, 827–848 (French). MR 860674, DOI 10.1215/S0012-7094-86-05345-7
John L. Harer, The cohomology of the moduli space of curves, Theory of moduli (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1337, Springer, Berlin, 1988, pp. 138–221. MR 963064, DOI 10.1007/BFb0082808
J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), no. 3, 457–485. MR 848681, DOI 10.1007/BF01390325
Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia, A characterization of arithmetic subgroups of $\textrm {SL}(2,\textbf {R})$ and $\textrm {SL}(2,\textbf {C})$, Math. Nachr. 159 (1992), 245–270. MR 1237113, DOI 10.1002/mana.19921590117
E. Hlawka, 90 Jahre Geometrie der Zahlen, Yearbook: surveys of mathematics 1980 (German), Bibliographisches Inst., Mannheim, 1980, pp. 9–41 (German). MR 620325
H. Huber. Ueber eine neue Klasse automorpher Formen und ein Gitterproblem in der hyperbolischen Ebene I. Comment. Math. Helv. 30 (1955), 20-62.
Heinz Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann. 138 (1959), 1–26 (German). MR 109212, DOI 10.1007/BF01369663
Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
N. V. Ivanov, Complexes of curves and Teichmüller modular groups, Uspekhi Mat. Nauk 42 (1987), no. 3(255), 49–91, 255 (Russian). MR 896878
Henryk Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136–159. MR 743969, DOI 10.1515/crll.1984.349.136
Henryk Iwaniec, Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matemática Iberoamericana. [Library of the Revista Matemática Iberoamericana], Revista Matemática Iberoamericana, Madrid, 1995. MR 1325466
F.W. Jenny. Ueber den ersten Eigenwert des Laplace-Operators auf einer Schar kompakter Riemannscher Flächen. Comment. Math. Helv. 59 (1984), 193-203.
Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
Steven P. Kerckhoff, Lines of minima in Teichmüller space, Duke Math. J. 65 (1992), no. 2, 187–213. MR 1150583, DOI 10.1215/S0012-7094-92-06507-0
Stefan Kühnlein, Partial solution of a conjecture of Schmutz, Arch. Math. (Basel) 67 (1996), no. 2, 164–172. MR 1399834, DOI 10.1007/BF01268932
J. Lehner and M. Sheingorn, Simple closed geodesics on $H^{+}/\Gamma (3)$ arise from the Markov spectrum, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 359–362. MR 752798, DOI 10.1090/S0273-0979-1984-15307-2
Feng Luo, Geodesic length functions and Teichmüller spaces, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 1, 34–41. MR 1405967, DOI 10.1090/S1079-6762-96-00008-X
W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces, Comm. Math. Phys. 161 (1994), no. 2, 419–432. MR 1266491, DOI 10.1007/BF02099785
Wen Zhi Luo and Peter Sarnak, Quantum ergodicity of eigenfunctions on $\textrm {PSL}_2(\mathbf Z)\backslash \mathbf H^2$, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 207–237. MR 1361757
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
J. Marklof, On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds, Nonlinearity 9 (1996), no. 2, 517–536. MR 1384490, DOI 10.1088/0951-7715/9/2/014
A. Markoff. Sur les formes binaires indéfinies. Math. Annalen 15 (1879), 381-406.
G. McShane. A remarkable identity for lengths of curves. PhD Thesis, University of Warwick (1991).
G. McShane. Simple geodesics and a series constant over Teichmüller space. To appear in Invent. math.
Greg McShane and Igor Rivin, Simple curves on hyperbolic tori, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 12, 1523–1528 (English, with English and French summaries). MR 1340065
Greg McShane and Igor Rivin, A norm on homology of surfaces and counting simple geodesics, Internat. Math. Res. Notices 2 (1995), 61–69. MR 1317643, DOI 10.1155/S1073792895000055
H. Minkowski. Geometrie der Zahlen. Teubner 1896/1910; Chelsea (1953).
Rick Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. MR 1326604, DOI 10.1090/gsm/005
Marston Morse, Topologically non-degenerate functions on a compact $n$-manifold $M$, J. Analyse Math. 7 (1959), 189–208. MR 113233, DOI 10.1007/BF02787685
David Mumford, A remark on Mahler’s compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294. MR 276410, DOI 10.1090/S0002-9939-1971-0276410-4
Jakob Nielsen, Jakob Nielsen: collected mathematical papers. Vol. 1, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1986. Edited and with a preface by Vagn Lundsgaard Hansen. MR 865335
B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148–211. MR 960228, DOI 10.1016/0022-1236(88)90070-5
B. Osgood, R. Phillips, and P. Sarnak, Moduli space, heights and isospectral sets of plane domains, Ann. of Math. (2) 129 (1989), no. 2, 293–362. MR 986795, DOI 10.2307/1971449
R. C. Penner, Perturbative series and the moduli space of Riemann surfaces, J. Differential Geom. 27 (1988), no. 1, 35–53. MR 918455, DOI 10.4310/jdg/1214441648
R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770, DOI 10.1515/9781400882458
Andrew D. Pollington and William Moran (eds.), Number theory with an emphasis on the Markoff spectrum, Lecture Notes in Pure and Applied Mathematics, vol. 147, Marcel Dekker, Inc., New York, 1993. MR 1219317
J. R. Quine and Peter Sarnak (eds.), Extremal Riemann surfaces, Contemporary Mathematics, vol. 201, American Mathematical Society, Providence, RI, 1997. Papers from the AMS Special Session held at the Annual Meeting of the American Mathematical Society in San Francisco, CA, January 4–5, 1995. MR 1429188, DOI 10.1090/conm/201
Burton Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity, Proc. Amer. Math. Soc. 78 (1980), no. 3, 455–456. MR 553396, DOI 10.1090/S0002-9939-1980-0553396-1
B. Riemann. Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. In Gesammelte Werke. Collected papers. Springer + Teubner, Leipzig (1990), 35-75.
I. Rivin. Private communication.
P. Sarnak. Prime geodesic theorems. Doctoral thesis, Stanford (1980).
Peter Sarnak, Extremal geometries, Extremal Riemann surfaces (San Francisco, CA, 1995) Contemp. Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 1–7. MR 1429189, DOI 10.1090/conm/201/02616
Paul Schmutz, Congruence subgroups and maximal Riemann surfaces, J. Geom. Anal. 4 (1994), no. 2, 207–218. MR 1277506, DOI 10.1007/BF02921547
Paul Schmutz, Systoles on Riemann surfaces, Manuscripta Math. 85 (1994), no. 3-4, 429–447. MR 1305753, DOI 10.1007/BF02568209
Paul Schmutz, Systoles on Riemann surfaces, Manuscripta Math. 85 (1994), no. 3-4, 429–447. MR 1305753, DOI 10.1007/BF02568209
Paul Schmutz, New results concerning the number of small eigenvalues on Riemann surfaces, J. Reine Angew. Math. 471 (1996), 201–220. MR 1374923, DOI 10.1515/crll.1996.471.201
Paul Schmutz, Arithmetic groups and the length spectrum of Riemann surfaces, Duke Math. J. 84 (1996), no. 1, 199–215. MR 1394753, DOI 10.1215/S0012-7094-96-08407-0
P. Schmutz Schaller, The modular torus has maximal length spectrum, Geom. Funct. Anal. 6 (1996), no. 6, 1057–1073. MR 1421874, DOI 10.1007/BF02246996
Paul Schmutz Schaller, Extremal Riemann surfaces with a large number of systoles, Extremal Riemann surfaces (San Francisco, CA, 1995) Contemp. Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 9–19. MR 1429190, DOI 10.1090/conm/201/02617
P. Schmutz Schaller. Geometric characterization of hyperelliptic Riemann surfaces. Preprint (1996).
P. Schmutz Schaller. Systole is a topological Morse function for Riemann surfaces. Preprint (1997).
P. Schmutz Schaller. Teichmüller space and fundamental domains of Fuchsian groups. Preprint (1997).
R. Schoen, S. Wolpert, and S. T. Yau, Geometric bounds on the low eigenvalues of a compact surface, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 279–285. MR 573440
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Mika Seppälä and Tuomas Sorvali, Geometry of Riemann surfaces and Teichmüller spaces, North-Holland Mathematics Studies, vol. 169, North-Holland Publishing Co., Amsterdam, 1992. MR 1202043
C.L. Siegel. Ueber die Klassenzahl quadratischer Zahlkörper. Acta Arith. 1 (1935), 83-86.
Carl Ludwig Siegel, Lectures on the geometry of numbers, Springer-Verlag, Berlin, 1989. Notes by B. Friedman; Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter; With a preface by Chandrasekharan. MR 1020761, DOI 10.1007/978-3-662-08287-4
Kisao Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), no. 4, 600–612. MR 398991, DOI 10.2969/jmsj/02740600
William P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 91–112. MR 903860
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
Anthony J. Tromba, Teichmüller theory in Riemannian geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. Lecture notes prepared by Jochen Denzler. MR 1164870, DOI 10.1007/978-3-0348-8613-0
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
A. B. Venkov, Spectral theory of automorphic functions, Trudy Mat. Inst. Steklov. 153 (1981), 172 (Russian). MR 665585
G. Voronoï. Sur quelques propriétés des formes quadratiques positives parfaites. J. reine angew. Math. 133 (1908), 97-178.
Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479. MR 982185
Scott Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), no. 4, 969–997. MR 796909, DOI 10.2307/2374363
Kazuo Azukawa, The invariant pseudometric related to negative plurisubharmonic functions, Kodai Math. J. 10 (1987), no. 1, 83–92. MR 879385, DOI 10.2996/kmj/1138037363
M. M. Elin, Epimorphism of the convolution operator in convex domains in $\textbf {C}^n$, Sibirsk. Mat. Zh. 27 (1986), no. 4, 201–203, 216 (Russian). MR 867872
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4