Classification of Stokes graphs of second order Fuchsian differential equations of genus two (Q2467703)

Stokes graphs, that is, graphs consisting of turning points, (regular) singular points and Stokes curves, play an important role in the global analysis of second order Fuchsian differential equations with a large parameter by means of the exact (complex) WKB analysis. When Fuchsian equations have three or four regular singular points (i.e., in the case when the Riemann surface of the square root of the potential is of genus zero or one), topological classification of Stokes graphs and global analysis of equations were explicitly done in \textit{M. Sato}, \textit{T. Aoki}, \textit{T. Kawai} and \textit{Y. Takei} [RIMS Koukyuuroku, No. 750, 43--51 (1991)] (see also \textit{T. Kawai} and \textit{Y. Takei} [Algebraic Analysis of Singular Perturbation Theory (Translation of Mathematical Monographs 227, AMS) (2005; Zbl 1100.34004)]). In this paper a complete list of topological patterns of Stokes graphs is given for Fuchsian equations with five regular singular points (i.e., when the Riemann surface of the square root of the potential is of genus two); it contains 25 different patterns. To classify all possible configurations the authors use (i) the connection between Stokes graphs and triangulations of a special kind of the Riemann sphere that was observed in the above references, and (ii) the two procedures for such triangulations, namely, reduction and blow up, which are first introduced in this paper.