## 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

### Citations:

Zbl 0515.53037; Zbl 1082.53033
Full Text:

### References:

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