Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces

Christian Klein, Alexey Kokotov, and Dmitry Korotkin

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Submission date: 11. Oct. 2005 (revised version: March 2006)
Pages: 39
published in: Mathematische Zeitschrift, 261 (2009) 1, p. 73-108 
DOI number (of the published article): 10.1007/s00209-008-0314-9
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Abstract:
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 indetify 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 three of Bolza's strata of symmetric Riemann surfaces of genus two.