Fixed point free involutions on Riemann surfaces. (English) Zbl 1157.30031

Parlier, Hugo

Isr. J. Math. 166, 297-311 (2008).

Let \(S\) be an orientable surface

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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.

Reviewer: Gou Nakamura (Toyota)

MSC:

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

Keywords:

hyperbolic Riemann surface; hyperelliptic Riemann surface; involution; hyperbolic metric; systole

Citations:

Zbl 0515.53037; Zbl 1082.53033

References:

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