The Bers's constant in genus two
This article deals with compact Riemann surfaces of genus \(g\geq 2\) endowed with their hyperbolic (Poincaré) metric. Lipman Bers has observed that each such Riemann surface has a family of \(3g-3\) simple and closed geodesics. Their length can be bounded by a certain constant, called the Bers constant \(B(g)\), depending on the genus \(g\). In the article it is shown that \(B(2)\) is given by \(\cosh B(2)/12=x_0\), where \(x_0\) is the unique real solution of the equation \(32x^5-32x^4-24x^3+24x^2-1=0\) which is larger than 1. The value of \(x_0\) is roughly 4.67. The surface which realises \(B(2)\) is essentially unique. The author introduces the contiguity graph, which measures in some sense the repartitions of the Weierstrass points on the surface, as a tool and uses it for the proof.
scientific article; zbMATH DE number 5944174