Quadratic Differentials with Closed Trajectories
Quadratic differentials with closed trajectories were first considered by Teichmüller in his “Habilitationsschrift”  in the following example: two non overlapping punctured disks G′ and G″, with punctures at z =0 and z = ∞ respectively, which do not contain z = −1, have to be chosen in such a way that their reduced moduli M′ and M″ maximize the sum q 2 M′ + M″. The solution (“Extremalgebiete des speziellen Modulsatzes”, pg 33) is given by a quadratic differential with second order poles at 0 and ∞. For Riemann surfaces and finitely many punctures, the problem was solved by the author in  and , see §23 below.
KeywordsLime Dinates Summing
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