Variation of Bergman metrics on Riemann surfaces

We study the variation of the Berman metric K(t,)|d|² on the Riemann surface R(t) (which varies with complex parameter t in a disk B) such that :B is a 2-dimensional Stein space with ^–1(t)=R(t),t B. We show that log K(t,)|d|² is plurisubharmonic in , and apply it to give the uniformization condition for the special Stein space.Revised version: 18 February 2004Mathematics Subject Classification (2000): 30C85, 30F15, 32F05, 32H10     

Foliations of asymptotically flat manifolds by surfaces of Willmore type

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch–Hawking mass. Thus our result has applications in the theory of general relativity.     

Zeta-determinant and torsion functions on Riemann surfaces of finite volume

We define ζ-determinant andL ²-analytic torsion functions for a Riemann surface of finite volume. We use the Selberg trace formula to express these determinant and torsion functions in terms of four Zeta functions which are related to the structure of discrete groups. A new invariant is also obtained.     

The argument principle and holomorphic extendibility to finite Riemann surfaces

Let M be a finite Riemann surface and let A(M) be the algebra of all continuous functions on M∪bM which are holomorphic on M. We prove that a continuous function Φ on bM extends to a function in A(M) if and only if for every f,g in A(M) such that fΦ+g≠0 on bM, the change of argument of fΦ+g is nonnegative.     

An isometry theorem for quadratic differentials on Riemann surfaces of finite genus

Assume both and are Riemann surfaces which are subsets of compact Riemann surfaces and respectively, and that the set has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on and are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of onto the Teichmüller space of is induced by some quasiconformal map of onto . Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.

     

Multivariate interpolation with increasingly flat radial basis functions of finite smoothness

In this paper, we consider multivariate interpolation with radial basis functions of finite smoothness. In particular, we show that interpolants by radial basis functions in ℝ ^d with finite smoothness of even order converge to a polyharmonic spline interpolant as the scale parameter of the radial basis functions goes to zero, i.e., the radial basis functions become increasingly flat.     

On the ergodicity of flat surfaces of finite area

We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmüller orbits are recurrent to a compact set of ${SL(2,\mathbb{R})/SL(S,\alpha)}$ , where SL(S,α) is the Veech group of the surface. In this setting, this means that the translation flow on a flat surface can be renormalized through its Veech group. This result applies in particular to flat surfaces of infinite genus and finite area. Our second result is an criterion for ergodicity based on the control of deforming metric of a flat surface. Applied to translation flows on compact surfaces, it improves and generalizes a theorem of Cheung and Eskin et al. (Partially hyperbolic dynamics, laminations, and Teichmüller flow, Fields Inst. Commun., Vol. 51. Amer. Math. Soc., Providence, pp. 213–221, 2007).     

Casimir effects on Riemann surfaces

The Casimir effect whose existence was first predicted by Casimir in 1948 is considered as a manifestation of macroscopic quantum field theory. This force is evaluated theoretically by using the value of the Riemann zeta function at −3. The aim of the present paper is to introduce a similar Casimir energy for a Riemann surface, and to express it by a special value of the Mellin transform of a theta series arising from the heat kernel and also by a weighted integral of the logarithm of the Selberg zeta function.     

Harmonic spinors on Riemann surfaces

We calculate the dimension of the space of harmonic spinors on hyperelliptic Riemann surfaces for all spin structures. Furthermore, we present non-hype relliptic examples of genus 4 and 6 on which the maximal possible number of linearly independent harmonic spinors is achieved.The second author was supported by the Schweizerischer Nationalfonds zur Förderung wissenschaftlicher Forschung     

p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.