zbMATH — the first resource for mathematics

Fixed point free involutions on Riemann surfaces. (English) Zbl 1157.30031
Let \(S\) be an orientable surface of even genus with a Riemannian metric \(d\) and with a fixed point free, orientation reversing involution \(\tau\). Then it is conjectured that there exists a point \(p\in S\) satisfying
\[ \frac{d(p,\tau(p))^2}{\text{area}(S)}\leq \frac{\pi}{4}. \]
This conjecture originated from the filling area conjecture by M. Gromov [J. Differ. Geom. 18, 1–147 (1983; Zbl 0515.53037)]. In the case that \(S\) is hyperelliptic, it was positively solved by V. Bangert, C. Croke, S. Ivanov, and M. Katz [Geom. Funct. Anal. 15, No. 3, 577–597 (2005; Zbl 1082.53033)]. The situation is different when \(S\) has odd genus. One of the main results in this paper is that for any odd \(g\geq 3\) and positive constant \(k\), there exists a hyperbolic Riemann surface \(S\) of genus \(g\) with an orientation reversing involution \(\tau\) such that \(d(p,\tau(p))>k\) holds for all \(p\in S\). This result is true in the case that \(\tau\) is an orientation preserving involution. The other main result concerns the sharp bound for hyperbolic metrics in genus 2 surfaces, that is, for a Riemann surface \(S\) of genus 2 with a hyperbolic metric and with an involution \(\tau\), there exists a point \(p\in S\) satisfying \(d(p,\tau(p))\leq \text{arccosh} \frac{5+\sqrt{17}}{2}\). It is mentioned that the surface which attains the sharp bound is not in the conformal class of the Bolza curve.

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
30F50 Klein surfaces
PDF BibTeX Cite
Full Text: DOI arXiv
[1] V. Bangert, C. Croke, S. Ivanov, and M. Katz, Filling area conjecture and ovalless real hyperelliptic surfaces, Geometric and Functional Analysis 15 (2005), 577–597. · Zbl 1082.53033
[2] P. Buser, The collar theorem and examples, Manuscripta Mathematica 25 (1978), 349–357. · Zbl 0402.53028
[3] P. Buser, Geometry and spectra of compact Riemann surfaces, of Progress in Mathematics, vol. 106, Birkhäuser Boston Inc., Boston, MA, 1992. · Zbl 0770.53001
[4] M. Gendulphe, Paysage systolique des surfaces hyperboliques compactes de caracteristique-1, available on the arxiv:math.DG/0508036, 2005.
[5] M. Gromov, Filling Riemannian manifolds, Journal of Differential Geometry 18 (1983), 1–147. · Zbl 0515.53037
[6] S. V. Ivanov and M. G. Katz. Generalized degree and optimal Loewner-type inequalities, Israel Journal of Mathematics 141 (2004), 221–233. · Zbl 1067.53031
[7] M. G. Katz and S. Sabourau. An optimal systolic inequality for CAT(0) metrics in genus two, Pacific Journal of Mathematics 227 (2006), 95–107. · Zbl 1156.53019
[8] L. Keen, Collars on Riemann surfaces, in Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Annals of Mathematics Studies, Princeton University Press, Princeton, N.J., 1974, pp. 263–268.
[9] I. Kra and B. Maskit. Bases for quadratic differentials, Commentarii Mathematici Helvetici 57 (1982), 603–626. · Zbl 0527.30036
[10] B. Randol, Cylinders in Riemann surfaces, Commentarii Mathematici Helvetici 54 (1979), 1–5. · Zbl 0401.30036
[11] P. Schmutz. Riemann surfaces with shortest geodesic of maximal length, Geometric and Functional Analysis 3 (1993), 564–631. · Zbl 0810.53034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.