6533b829fe1ef96bd128ad47

RESEARCH PRODUCT

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

Christian KleinChristian KleinD. KorotkinAlexey Kokotov

subject

Mathematics(all)General MathematicsRiemann surface010102 general mathematicsMathematical analysis01 natural sciencesModuli spaceRiemann–Hurwitz formulaModuli of algebraic curvesRiemann Xi functionMathematics - Spectral Theorysymbols.namesakeRiemann problemMathematics::Algebraic GeometryGenus (mathematics)0103 physical sciencesFOS: Mathematicssymbols14H15010307 mathematical physics0101 mathematicsSpectral Theory (math.SP)Bergman metricMathematics

description

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 orbifold Euler characteristics of the moduli space. A similar analysis is performed for the Bolza's strata of symmetric Riemann surfaces of genus two.

yearjournalcountryeditionlanguage
2005-11-08Mathematische Zeitschrift
10.1007/s00209-008-0314-9http://dx.doi.org/10.1007/s00209-008-0314-9