Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hyperelliptic surfaces are Loewner

Authors: Mikhail G. Katz and Stéphane Sabourau
Journal: Proc. Amer. Math. Soc. 134 (2006), 1189-1195
MSC (2000): Primary 53C23; Secondary 30F10
Published electronically: July 20, 2005
MathSciNet review: 2196056
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that C. Loewner's inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces $X$ as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to $X$ away from Weierstrass points. The loops are then transplanted to $X$, and surgered to obtain a Loewner loop on $X$. In higher genus, we exploit M. Gromov's area estimates for $\varepsilon$-regular metrics on $X$.

References [Enhancements On Off] (What's this?)

  • [Bab93] I. K. Babenko, Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 707–751 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 1–38. MR 1208148, 10.1070/IM1993v041n01ABEH002181
  • [Bal04] Balacheff, F.: Sur des problèmes de la géométrie systolique. Sémin. Théor. Spectr. Géom. Grenoble, 22 (2004), 71-82.
  • [BCIK04] Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Boundary case of equality in optimal Loewner-type inequalities, Trans. A.M.S., to appear. See arXiv:math.DG/0406008
  • [BCIK05] Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Filling area conjecture and ovalless real hyperelliptic surfaces, Geometric and Functional Analysis (GAFA) 15 (2005), no. 3. See arXiv:math.DG/0405583
  • [BK03] Victor Bangert and Mikhail Katz, Stable systolic inequalities and cohomology products, Comm. Pure Appl. Math. 56 (2003), no. 7, 979–997. Dedicated to the memory of Jürgen K. Moser. MR 1990484, 10.1002/cpa.10082
  • [BK04] Bangert, V; Katz, M.: An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm. Comm. Anal. Geom. 12 (2004), number 3, 703-732. See arXiv:math.DG/0304494
  • [CK03] Christopher B. Croke and Mikhail Katz, Universal volume bounds in Riemannian manifolds, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 109–137. MR 2039987, 10.4310/SDG.2003.v8.n1.a4
  • [FK92] H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765
  • [Gr83] Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
  • [Gr96] Mikhael Gromov, Systoles and intersystolic inequalities, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 291–362 (English, with English and French summaries). MR 1427763
  • [Gr99] Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320
  • [He81] James J. Hebda, Some lower bounds for the area of surfaces, Invent. Math. 65 (1981/82), no. 3, 485–490. MR 643566, 10.1007/BF01396632
  • [IK04] Sergei V. Ivanov and Mikhail G. Katz, Generalized degree and optimal Loewner-type inequalities, Israel J. Math. 141 (2004), 221–233. MR 2063034, 10.1007/BF02772220
  • [Ka03] Mikhail Katz, Four-manifold systoles and surjectivity of period map, Comment. Math. Helv. 78 (2003), no. 4, 772–786. MR 2016695, 10.1007/s00014-003-0774-9
  • [KL04] Katz, M.; Lescop, C.: Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers, Proceedings of conference and workshop in memory of R. Brooks, Israel Mathematical Conference Proceedings, Contemporary Math., A.M.S., Providence, R.I. (to appear).
  • [KR04] Katz, M.; Rudyak, Y.: Lusternik-Schnirelmann category and systolic category of low dimensional manifolds. Communications on Pure and Applied Mathematics, to appear. See arXiv:math.DG/0410456
  • [KS04] Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory and Dynamical Systems, 25 (2005). See arXiv:math.DG/0410312
  • [KS05] Katz, M.; Sabourau, S.: An optimal systolic inequality for CAT(0) metrics in genus two, preprint. See arXiv:math.DG/0501017
  • [Kod87] Shigeru Kodani, On two-dimensional isosystolic inequalities, Kodai Math. J. 10 (1987), no. 3, 314–327. MR 929991, 10.2996/kmj/1138037462
  • [Kon03] Jian Kong, Seshadri constants on Jacobian of curves, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3175–3180 (electronic). MR 1974680, 10.1090/S0002-9947-03-03305-1
  • [Ku78] H. T. Kung and Charles E. Leiserson, Systolic arrays (for VLSI), Sparse Matrix Proceedings 1978 (Sympos. Sparse Matrix Comput., Knoxville, Tenn., 1978) SIAM, Philadelphia, Pa., 1979, pp. 256–282. MR 566379
  • [LLS90] J. C. Lagarias, H. W. Lenstra Jr., and C.-P. Schnorr, Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice, Combinatorica 10 (1990), no. 4, 333–348. MR 1099248, 10.1007/BF02128669
  • [Mi95] Rick Miranda, Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. MR 1326604
  • [Pu52] P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71. MR 0048886
  • [Sa04] Stéphane Sabourau, Systoles des surfaces plates singulières de genre deux, Math. Z. 247 (2004), no. 4, 693–709 (French, with English summary). MR 2077416, 10.1007/s00209-003-0641-9
  • [Sa05] Sabourau, S.: Systolic volume and minimal entropy of aspherical manifolds, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C23, 30F10

Retrieve articles in all journals with MSC (2000): 53C23, 30F10

Additional Information

Mikhail G. Katz
Affiliation: Department of Mathematics and Statistics, Bar Ilan University, Ramat Gan 52900, Israel

Stéphane Sabourau
Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37400 Tours, France
Address at time of publication: Mathematics and Computer Science Department, St. Joseph’s University, 5600 City Avenue, Philadelphia, Pennsylvania 19131

Keywords: $\varepsilon$-regular metrics, Hermite constant, hyperelliptic involution, Loewner inequality, Pu's inequality, systole, Weierstrass point
Received by editor(s): March 18, 2004
Received by editor(s) in revised form: October 26, 2004
Published electronically: July 20, 2005
Additional Notes: The first author was supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03)
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2005 American Mathematical Society