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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.

##### MSC:
 53C22 Geodesics in global differential geometry 30F60 Teichmüller theory for Riemann surfaces
##### Keywords:
Teichmüller space; systole; maximal surfaces
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