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An inequality for Riemann surfaces. (English) Zbl 0575.30039
Differential geometry and complex analysis, Vol. dedic. H. E. Rauch, 87-93 (1985).
[For the entire collection see Zbl 0561.00010.]
Let S be a Riemann surface of type (p,n), (p-the genus, \(n=\nu +\mu\), \(\nu\)-the number of punctures and \(\mu\)-that of the ideal bounday curves) with \(a=2p-2+n>0\). A loop is a simple closed (Poincaré) geodesic on S and is called outer or inner as it can be deformed into an ideal boundary curve of S or not. One proves that 1) the length \(\ell\) of the shortest inner loop (if any) of S is bounded from above by a bound depending only on a and L (the length of the longest outer loop) (if any), and moreover, that 2) there are \(d=3p-3+n\) disjoint inner loops on S whose length do not exceed a bound like that in 1). Besides this inequality of importance in the theory of moduli, six lemmas give useful geometric information on loops and collars, and estimates on lengths and distances.
Reviewer: C.Andreian Cazacu

MSC:
30F10 Compact Riemann surfaces and uniformization
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)