Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Consider the moduli space of pairs (C,) where C is a smooth compact complex curve of a given genus and is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.     

An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces

An explicit upper bound for the Weil-Petersson volumes of punctured Riemann surfaces is obtained using Penner's combinatorial integration scheme from [4]. It is shown that for a fixed number of punctures n and for genus g increasing, while this limit is exactly equal to two for n=1. Received: 17 May 2000 / Revised version: 9 August 2000 / Published online: 23 July 2001     

Brownian motion and the formation of singularities in the heat flow for harmonic maps

Summary We develop a general framework for a stochastic interpretation of certain nonlinear PDEs on manifolds. The linear operation of takin expectations is replaced by the concept of martingale means, namely the notion of deterministic starting points of martingales (with respect to the Levi-Civita connection) ending up at a prescribed state. We formulate a monotonicity condition for the Riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation. Such estimates lead to simple criteria for blow-ups in the nonlinear heat flow for harmonic maps with small initial energy.This article was processed by the author using the Springer-Verlag TEX QPMZGHB macro package 1991.     

On quadratic functionals of the Brownian sheet and related processes

Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on [0,1]²$[0,1{]}^2$. In particular: (i) we use Fubini-type techniques to establish identities in law with quadratic functionals of other Gaussian processes, (ii) we explicitly calculate the Laplace transform of such functionals by means of Karhunen–Loève expansions, (iii) we prove central and non-central limit theorems in the spirit of Peccati and Yor [Four limit theorems involving quadratic functionals of Brownian motion and Brownian bridge, Asymptotic Methods in Stochastics, American Mathematical Society, Fields Institute Communication Series, 2004, pp. 75–87] and Nualart and Peccati [Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33(1) (2005) 177–193]. Our results extend some classical computations due to Lévy [Wiener's random function and other Laplacian random functions, in: Second Berkeley Symposium in Probability and Statistics, 1950, pp. 171–186], as well as the formulae recently obtained by Deheuvels and Martynov [Karhunen–Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, Progress in Probability, vol. 55, Birkhäuser Verlag, Basel, 2003, pp. 57–93].     

Meromorphic quadratic differentials with prescribed singularities

We study the singular flat structure associated to any meromorphic quadratic differential on a compact Riemann surface to prove an existence theorem as follows. There exists a meromorphic quadratic differential with given orders of the poles and zeros and orientability or non orientability of the horizontal foliation, iff these prescribed topological data are admissible according to the Gauss-Bonnet Theorem, the Residue Theorem and certain conditions arising from local orientability or non orientablity considerations. Some few exceptional cases remain excluded. Thus, we generalize two previous results. One due to Masur & Smillie, which assumes that poles are at most simple; and a second one due to Muciño-Raymundo, which assumes that the horizontal foliation is orientable.Partially supported by DGAPA-UNAM and CONACYT 28492-E.     

The eventually distance minimizing rays in moduli spaces

The eventually distance minimizing ray(EDM ray) in moduli spaces of the Riemann surfaces of analytic finite type with 3 g + n-3 0 is studied, which was introduced by Farb and Masur [5]. The asymptotic distance of EDM rays in a moduli space and the distance of end points of EDM rays in the boundary of the moduli space in the augmented moduli space are discussed in this article. A relation between the asymptotic distance of EDM rays and the distance of their end points is established. It is proved also that the distance of end points of two EDM rays is equal to that of end points of two Strebel rays in the Teichmu¨ller space of a covering Riemann surface which are leftings of some representatives of the EDM rays. Meanwhile, simpler proofs for some known results are given.     

Genus one polyhedral surfaces,spaces of quadratic differentials on tori and determinants of Laplacians

We prove a formula for the determinant of the Laplacian on an arbitrary compact polyhedral surface of genus one. This formula generalizes the well-known Ray–Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic surface is also considered. We study the determinant of the Laplacian as a functional on the moduli space of meromorphic quadratic differentials with L simple poles and L simple zeros and derive formulas for variations of this functional with respect to natural coordinates on . We give also a new proof of Troyanov’s theorem stating the existence of a conformal flat conical metric on a compact Riemann surface of arbitrary genus with a prescribed divisor of conical points.     

Functional limit theorems for generalized quadratic variations of Gaussian processes

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).     

Difféomorphismes holomorphes Anosov

We classify the holomorphic diffeomorphisms of complex projective varieties with an Anosov dynamics and holomorphic stable and unstable foliations: The variety is finitely covered by a compact complex torus and the diffeomorphism corresponds to a linear transformation of this torus.

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Difféomorphismes holomorphes Anosov

     

Logarithmic moduli spaces for surfaces of class VII

In this paper we describe logarithmic moduli spaces of pairs (S, D) consisting of a minimal surface S of class VII with second Betti number b ₂ > 0 together with a reduced maximal divisor D of b ₂ rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of S\D and for the associated holomorphic contraction . Part of this work was done while the first author visited the University of Osnabrück under the program “Globale Methoden in der komplexen Geometrie” of the DFG and while the second author visited the Max-Planck-Institut für Mathematik in Bonn and the LATP, Université de Provence. We thank these institutions for their hospitality and for financial support. Furthermore the authors wish to thank Georges Dloussky for numerous discussions on surfaces of class VII.     

Deflating quadratic matrix polynomials with structure preserving transformations

Given a pair of distinct eigenvalues (λ₁,λ₂) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form having the same eigenvalue s as Q(λ), with Q_d(λ) an (n-1)×(n-1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ₁ and λ₂. This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(λ) unless the eigenvectors corresponding to λ₁ and λ₂ are parallel. We identify conditions under which we can construct a family of 2n×2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(λ), (c) preserve the block structure of a large class of block symmetric linearizations of Q(λ), thereby defining new quadratic matrix polynomials Q₁(λ) that have the same eigenvalue s as Q(λ), (d) yield quadratics Q₁(λ) with the property that their eigenvectors associated with λ₁ and λ₂ are parallel and hence can subsequently be deflated by a similarity applied directly to Q₁(λ). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.     

Generalized bochner theorems and the spectrum of complete manifolds

Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L ² harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L ² harmonic spinors on spin manifolds.     

Harbater-Mumford subvarieties of moduli spaces of covers

We define Harbater-Mumford subvarieties, which are special kinds of closed subvarieties of Hurwitz moduli spaces obtained by fixing some of the branch points. We show that, for many finite groups, finding geometrically irreducible HM-subvarieties defined over is always possible. This provides information on the arithmetic of Hurwitz spaces and applies in particular to the regular inverse Galois problem with (almost all) fixed branch points. Profinite versions of our results can also be stated, providing new tools to study the geometry of modular towers and the regular inverse Galois problem for profinite groups.     

Grafting hyperbolic metrics and Eisenstein series

The family hyperbolic metric for the plumbing variety {zw = t} and the non holomorphic Eisenstein series are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of Riemann surfaces. Applications include an asymptotic expansion for the Weil–Petersson metric and a local form of symplectic reduction.     

Small ball probabilities of Gaussian fields

Summary Lower bounds on the small ball probability are given for Brownian sheet type Gaussian fields as well as for general Gaussian fields with stationary increments in ^d . In particular, a sharp bound is found for the fractional Lévy Brownian fields.The research is partly supported by a National University of Singapore's Research Project     

The harmonic measures of Lucy Garnett

This paper presents an improved approach to the theory of harmonic measures for foliated spaces introduced by Garnett. This approach is based on a method for solving elliptic equations on foliated spaces and on the Hille-Yosida theory. The diffusion semigroup of a general Laplacian and its infinitesimal generator are made explicit. Applications of the path space to the dynamical study of a foliated space are described. In particular, the final section studies cocycles on foliated spaces, a formula for their asymptotic limit, and some analytic and geometric consequences.     

Small values of Gaussian processes and functional laws of the iterated logarithm

Summary We estimate small ball probabilities for locally nondeterministic Gaussian processes with stationary increments, a class of processes that includes the fractional Brownian motions. These estimates are used to prove Chung type laws of the iterated logarithm.Research supported by the United States Air Force office of Scientific Research, Contract No. 91-0030