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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.

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
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References:
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