Closed geodesics on multiply connected manifolds

The existence of a finite number of non-hyperbolic periodic trajectories in Kirchhoff's problem of the motion of a rigid body in an ideal fluid as well as in its twin problem of the motion of a rigid body with a fixed point in a force field with a quadratic potential is proved using one of Klingenberg's theorems. The dynamical system is considered on a multiply connected manifold of even dimension with a Riemann metric.     

Schwarz-Christoffel mapping of multiply connected domains

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivitym≥2 thereby extending the results of Christoffel (1867) and Schwarz (1869) form=1 and Komatu (1945),m=2. A formula forf, the conformal map of the exterior ofm bounded disks to the exterior ofm bounded disjoint polygons, is derived. The derivation characterizes the global preSchwarzianf″ (z)/f′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on them boundary circles via the reflection principle and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a “method of images” infinite sequence of iterations of reflecting prevertex singularities from them boundary circles to the whole sphere.     

Subsonic irrotational flows in multiply connected domains

The solvability of a boundary value problem for a nonlinar elliptic partial differential equation in a plane, bounded, multiply connected domain is investigated. The solution of the problem is equal to unknown constants on inner components of the boundary. We want to determine these constants so that the solution satisfies the so-called trailing conditions. The results have immediate applications in hydrodynamics.     

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.     

Reproducing kernels for Hardy spaces on multiply connected domains

Associated with a boundedg-holed (g0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.     

Restricted T-universal functions on multiply connected domains

Let G ⊂ C be a domain which is bounded by a finite number of pairwise disjoint Jordan curves. We prove the existence of a function which is holomorphic exactly on G and has universal translates with respect to a prescribed set E ⊂∂G and which in addition is continuous on G ^-\E. This revised version was published online in June 2006 with corrections to the Cover Date.     

Green's functions for Laplace's equation in multiply connected domains

Analytical formulae for the first-type Green's function forLaplace's equation in multiply connected circular domains arepresented. The method is constructive and relies on the useof a special function known as the Schottky–Klein primefunction associated with multiply connected circular domains.It is shown that all the important functions of potential theory,including the first-type Green's function, the modified Green'sfunctions and the harmonic measures of a domain, can be writtenin terms of this prime function. A broad range of representativeexamples are given to demonstrate the efficacy of the methodas well as a quantitative comparison, in special cases, withthe results obtained using other independent methods.     

Theory of nonunivalent mappings of multiply connected domains

In this paper one investigates new classes of regular functions in arbitrary domains, both with one and two distinguished boundary components. Special cases of these classes are the classes of mappings considered by Yu. E. Alenitsyn, I. P. Mityuk, and T. Kubo.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 184–197, 1981.     

Isoperimetric inequalities in the torsion problem for multiply connected domains

Summary We construct upper bounds for warping function and torsional rigidity relative to degenerate elliptic matrices in multiply connected domains.

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Sunto Si ricavano limitazioni superiori per la funzione deformazione e la rigidità torsionale relative a matrici elittiche degeneri in domini a connessione multipla.

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Work performed under the auspices of G.N.A.F.A. of italian C.N.R.     

On multiply connected wandering domains of meromorphic functions

We describe conditions under which a multiply connected wanderingdomain of a transcendental meromorphic function with a finitenumber of poles must be a Baker wandering domain, and we discussthe possible eventual connectivity of Fatou components of transcendentalmeromorphic functions. We also show that if f is meromorphic,U is a bounded component of F(f) and V is the component of F(f)such that f(U)V, then f maps each component of U onto a componentof the boundary of V in . We give examples which show that our results are sharp; for example,we prove that a multiply connected wandering domain can mapto a simply connected wandering domain, and vice versa.     

On the approximate conformal mapping of multiply connected domains

Summary A method is presented for constructing approximations to the standard mappings for multiply connected regions given by Nehari [5]. The case of mapping onto a slit annulus is considered in detail, and computational results are presented for several examples.     

Analytic wavelets in multiply connected domains with circular boundaries

Wavelet bases in Hardy-type spaces of analytic functions in a domain with circular boundaries are constructed from a wavelet basis in the Hardy space of functions analytic in the disk. An estimate of the rate of convergence of the partial sums of the constructed wavelets is also obtained.     

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6.     

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.     

Interpolation problems and Toeplitz operators on multiply connected domains

LetR be a bounded domain in the complex plane bounded by n + 1 nonintersecting analytic Jordan curves, letE, F, andG be flat unitary vector bundles (in the sense of Abrahamse and Douglas) and let :F G and :E G be bounded analytic bundle maps. A condition is given for the existence of a bounded analytic map D:E F such that D = , together with an estimate for D. An interesting special case is the case whereE = G and = I _E , for which the condition involves a uniform lower bound for a class of Toeplitz operators overR, all of which are induced (formally) by the bundle map (N = rankE). When interpreted for a finite column of analytic scalar functions, this special case gives quantitative information on the corona theorem forR. The main tool is the Sz.Nagy-Foias commutant lifting theorem for regionsR recently obtained by the author.Research supported by National Science Foundation Grant No. MCS 77-00966.