Small eigenvalues of automorphic Laplacians in spaces of parabolic forms

In this paper, the Yang-Yau inequality for the first eigenvalue of the Laplace operator on a compact Riemann surface is carried to the case of Fuchsian groups of the first kind. With its help, for specific subgroups of the modular group PSL(2,), the existence of cuspidal representations of the complementary series in the decomposition of regular representations of the group PSL(2,) into irreducible components is proved. In addition, a lower bound for the degree of an arbitrary nonconstant meromorphic function, automorphic with respect to some congruence subgroup of PSL(2,) is given in terms of the index of in PSL (2,) only.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vo!o 134, pp. 157–168, 1984o     

Hyperbolic -spheres with conical singularities, accessory parameters and Kähler metrics on

We show that the real-valued function on the moduli space of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic -sphere with conical singularities of arbitrary orders , generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on parameterized by the set of orders , explicitly relate accessory parameters to these metrics, and prove that the functions are their Kähler potentials.

     

The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces

A local index theorem for families of -operators on Riemann surfaces with functures is proved. A new Kähler metric on the moduli space of punctured surfaces is described in terms of the Eisenstein-Maass series.     

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kähler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.     

Revealing Low-Radiative Modes of Nanoresonators with Internal Raman Scattering

JETP Letters - Revealing hidden non-radiative (dark) modes of resonant nanostructures using optical methods such as dark-field spectroscopy often becomes a sophisticated problem due to a weak...     

Influence of Continuous Wave Interference on the Efficiency of the Non-Threshold Search Procedure for a Noise-Like Signal by Delay Time with Transition to the Frequency Domain

Journal of Communications Technology and Electronics - A statistical study of the effectiveness of the non-threshold search procedure for a noise-like phase-shift keyed signal by the delay time is...     

A local index theorem for families of\bar \partial -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

We prove a local index theorem for families of \(\bar \partial \) -operators on Riemann surfaces of type (g, n), i.e. of genusg withn>0 punctures. We calculate the first Chern form of the determinant line bundle on the Teichmüller spaceT _g,n endowed with Quillen's metric (where the role of the determinant of the Laplace operators is played by the values of the Selberg zeta function at integer points). The result differs from the case of compact Riemann surfaces by an additional term, which turns out to be the Kähler form of a new Kähler metric on the moduli space of punctured Riemann surfaces. As a corollary of this result we derive, for instance, an analog of Mumford's isomorphism in the case of the universal curve.     

Isomonodromic tau function on the space of admissible covers

The isomonodromic tau function of the Fuchsian differential equations associated to Frobenius structures on Hurwitz spaces can be viewed as a section of a line bundle on the space of admissible covers. We study the asymptotic behavior of the tau function near the boundary of this space and compute its divisor. This yields an explicit formula for the pullback of the Hodge class to the space of admissible covers in terms of the classes of compactification divisors.