On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.     

Stochastic Loewner evolution in multiply connected domains

We construct radial stochastic Loewner evolution in multiply connected domains, choosing the unit disk with concentric circular slits as a family of standard domains. The natural driving function or input is a diffusion on the associated moduli space. The diffusion stops when it reaches the boundary of the moduli space. We show that for this driving function the family of random growing compacts has a phase transition for $\kappa =4$ and $\kappa =8$, and that it satisfies locality for $\kappa =6$. To cite this article: R.O. Bauer, R.M. Friedrich, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Real Quotient Singularities and Nonsingular Real Algebraic Curves in the Boundary of the Moduli Space

The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M_g/ of nonsingular real algebraic curves of genus g (g2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M_g/. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M_g/ is real analytic. We conclude that M_g/ is not a real analytic variety.     

Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed

We study a curvature-dependent motion of plane curves in a two-dimensional infinite cylinder with spatially undulating boundary. The law of motion is given by V=κ+A$V=\kappa +A$, where V is the normal velocity of the curve, κ is the curvature, and A is a positive constant. The boundary undulation is assumed to be almost periodic, or, more generally, recurrent in a certain sense. We first introduce the definition of recurrent traveling waves and establish a necessary and sufficient condition for the existence of such traveling waves. We then show that the traveling wave is asymptotically stable if it exists. Next we show that a regular traveling wave has a well-defined average speed if the boundary shape is strictly ergodic. Finally we study what we call “virtual pinning”, which means that the traveling wave propagates over the entire cylinder with zero average speed. Such a peculiar situation can occur only in non-periodic environments and never occurs if the boundary undulation is periodic.     

On an exterior boundary-value problem for the time-harmonic maxwell equations with boundary conditions for the normal components of the electric and magnetic field

We treat the time-harmonic Maxwell equations with the boundary condition (ν, E) = (ν, H) = 0 in an exterior multiply connected domain. A uniqueness result by Yee for the case of a simply connected domain is extended to multiply connected domains and existence is obtained by a boundary integral equation approach.     

The Douglas problem for parametric double integrals

 Let be a parametric variational double integral and γ ⊂ ℝ ⁿ be a system of several distinct Jordan curves. We prove the existence of multiply connected, conformally parametrized minimizers of spanned in γ by solving the Douglas problem for parametric functionals on multiply connected schlicht domains. As a by-product we obtain a simple isoperimetric inequality for multiply connected -minimizers, and we discuss regularity results up to the boundary which follow from corresponding results for the Plateau problem. Received: 19 April 2002 Mathematics Subject Classification (2000): 49J45, 49Q10, 53A07, 53A10     

Commutation relations for Schramm‐Loewner evolutions

Schramm‐Loewner evolutions (SLEs) describe a one‐parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e., several random curves). In particular, we derive infinitesimal commutation conditions, discuss some elementary solutions, study integrability conditions following from commutation, and show how to lift these infinitesimal relations to global relations in simple cases. The situation in multiply connected domains is also discussed. © 2007 Wiley Periodicals, Inc.     

The Carathéodory topology for multiply connected domains II

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.     

Schwarz problem for bi-analytic functions in multiply connected domains

The Schwarz problem for bi-analytic functions in unbounded circular multiply connected domains is considered. We combine constructive methods applied to boundary value problems for complex partial differential equations in simply connected domains and for the Riemann–Hilbert type problems in multiply connected domains. A general method is outlined and the case of doubly connected domains is discussed in details. Solution is obtained in the form of a series.     

Galois Covers of Moduli of Curves

Moduli spaces of pointed curves with some level structure are studied. We prove that for so-called geometric level structures, the levels encountered in the boundary are smooth if the ambient variety is smooth, and in some cases we can describe them explicitly. The smoothness implies that the moduli space of pointed curves (over any field) admits a smooth finite Galois cover. Finally, we prove that some of these moduli spaces are simply connected.     

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E→Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H^∗(BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces.We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.     

Exponential mixing for the Teichmüller flow

We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the $SL(2,\mathbb{R})We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of H?lder observables. A geometric consequence is that the action in the moduli space has a spectral gap.     

A moduli space of non-compact curves on a complex surface

We show that under mild boundary conditions the moduli space of non-compact curves on a complex surface is (locally) an analytic subset of a ball in a Banach manifold, defined by finitely many holomorphic functions.     

Period integrals,quantum numbers,and confinement in Susy QCD

We compute the period integrals on degenerate Seiberg—Witten curves for supersymmetric QCD explicitly and also show how these periods determine the changes in the quantum numbers of the states when passing from the weak- to strong-coupling domains in the mass moduli space of the theory. We discuss the confinement of monopoles at a strong coupling and demonstrate that the ambiguities in choosing the path in the moduli space do not affect the physical conclusions on confinement of monopoles in the phase with condensed light dyons.     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,II

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,I

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the scaling limit of loop-erased random walk excursion

We use the known convergence of loop-erased random walk to radial SLE(2) to give a new proof that the scaling limit of loop-erased random walk excursion in the upper half-plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs which is used together with a Beurling-type estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general simply connected domains.     

Functional-differential equations in a class of analytic functions and its application to elastic composites

According to Muskhelishvili’s approach, two-dimensional elastic problems for media with non-overlapping inclusions are reduced to boundary value problems for analytic functions in multiply connected domains. Using a method of functional equations developed by Mityushev, we reduce such a problem for a circular multiply connected domain to functional-differential equations. It is proved that the operator corresponding to the functional-differential equations is compact in the Hardy–Sobolev space. Moreover, these equations can be solved by the method of successive approximation under some natural conditions.     

Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe?s canonical slit domains

This paper presents a boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region G onto the second, third and fourth categories of Koebe?s canonical slit domains. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The method is an extension of the author?s method for the first category of Koebe?s canonical slit domains (see [M.M.S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput. 31 (2009) 1695-1715]). Three numerical examples are presented to illustrate the performance of the proposed method.