Riemann surfaces with shortest geodesic of maximal length. (English) Zbl 0810.53034

The following questions are treated: 1. For a fixed genus \(g\), the Riemann surface of constant curvature \(-1\) (called global maximal surface) is searched where the length of the systole, the shortest simple closed geodesic, is a global maximum in the Teichmüller space \(T_ g\). 2. All surfaces (called maximal surfaces) should be found where the length of the systole is a local maximum in \(T_ g\). The main result describing properties of these surfaces are the following: Theorem 1.1 A maximal surface of genus \(g\) has at least \(6g - 5\) shortest simple closed geodesics of equal length. Theorem 1.2. It is necessary and sufficient for a surface \(M\) of genus \(g\) to be a maximal surface: (i) \(M\) is strongly \(F\)-minimal; (ii) \(M\) is \(F\)-regular. Examples of such surfaces are given. So, maximal surfaces of signature \((1,n)\) and (2,2), maximal surfaces of genus 2 and maximal surfaces which are modelled upon Euclidean polyhedra are treated. Finally, two important maximal surfaces in genus 3 and 4 which may be global maximal surfaces are studied.


53C22 Geodesics in global differential geometry
30F60 Teichmüller theory for Riemann surfaces
Full Text: DOI EuDML


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