### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 058, 12 pages      arXiv:1801.08529      https://doi.org/10.3842/SIGMA.2018.058

### Fuchsian Equations with Three Non-Apparent Singularities

Alexandre Eremenko a and Vitaly Tarasov bc
a) Purdue University, West Lafayette, IN 47907, USA
b) Indiana University - Purdue University Indianapolis, Indianapolis, IN 46202, USA
c) St. Petersburg Branch of Steklov Mathematical Institute, St. Petersburg, 191023, Russia

Received February 02, 2018, in final form June 10, 2018; Published online June 15, 2018

Abstract
We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.

Key words: Fuchsian equations; hypergeometric equation; difference equations; apparent singularities; bispectral duality; positive curvature; conic singularities.

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References

1. Buckman R., Schmitt N., Spherical polygons and unitarization, unpublished, available at http://www.gang.umass.edu/reu/2002/gon.pdf.
2. Cui G., Gao Y., Rugh H.H., Tan L., Rational maps as Schwarzian primitives, Sci. China Math. 59 (2016), 1267-1284, arXiv:1511.04246.
3. Eremenko A., Metrics of positive curvature with conic singularities on the sphere, Proc. Amer. Math. Soc. 132 (2004), 3349-3355, math.MG/0208025.
4. Eremenko A., Co-axial monodromy, Ann. Sc. Norm. Super. Pisa, to appear, arXiv:1706.04608.
5. Eremenko A., Gabrielov A., Shapiro M., Vainshtein A., Rational functions and real Schubert calculus, Proc. Amer. Math. Soc. 134 (2006), 949-957, math.AG/0407408.
6. Eremenko A., Gabrielov A., Tarasov V., Metrics with conic singularities and spherical polygons, Illinois J. Math. 58 (2014), 739-755, arXiv:1405.1738.
7. Eremenko A., Gabrielov A., Tarasov V., Spherical quadrilaterals with three non-integer angles, J. Math. Phys. Anal. Geom. 12 (2016), 134-167, arXiv:1504.02928.
8. Fujimori S., Kawakami Y., Kokubu M., Rossman W., Umehara M., Yamada K., CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), 144-149, arXiv:1008.3734.
9. Heins M., On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1-60.
10. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
11. Klein F., Mathematisches Seminar zu Göttingen, Winter-Semester 1905/06, English transl. available at http://www.claymath.org/publications/klein-protokolle.
12. Mondello G., Panov D., Spherical metrics with conical singularities on a 2-sphere: angle constraints, Int. Math. Res. Not. 2016 (2016), 4937-4995, arXiv:1505.01994.
13. Mukhin E., Tarasov V., Varchenko A., Bispectral and $({\mathfrak{gl}}_N,{\mathfrak{gl}}_M)$ dualities, discrete versus differential, Adv. Math. 218 (2008), 216-265, math.QA/0605172.
14. Picard E., De l'intégration de l'équation $\Delta u=e^u$ sur une surface de Riemann fermée, J. Reine Angew. Math. 130 (1905), 243-258.
15. Scherbak I., Rational functions with prescribed critical points, Geom. Funct. Anal. 12 (2002), 1365-1380, math.QA/0205168.
16. Schilling F., Ueber die Theorie der symmetrischen $S$-Functionen mit einem einfachen Nebenpunkte, Math. Ann. 51 (1899), 481-522.
17. Shafarevich I.R., Basic algebraic geometry. I. Varieties in projective space, 2nd ed., Springer-Verlag, Berlin, 1994.
18. Troyanov M., Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), 793-821.