Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces. (English) Zbl 1210.58025
Summary: We study extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of compact genus two Riemann surfaces. By a combination of analytical and numerical methods, we identify four non-degenerate critical points of this function and compute the signature of the Hessian at these points. The curve with the maximal number of automorphisms (the Burnside curve) turns out to be the point of the absolute maximum. Our results agree with the mass formula for virtual Euler characteristics of the moduli space. A similar analysis is performed for Bolza’s strata of symmetric Riemann surfaces of genus two.
MSC:
| 58J52 | Determinants and determinant bundles, analytic torsion |
| 14H15 | Families, moduli of curves (analytic) |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
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