Browsing by Author "Radnell, David"

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  • Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces
    (2016-06-14) Radnell, David; Schippers, Eric; Staubach, Wolfgang
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    Consider a Riemann surface of genus (Formula presented.) bordered by (Formula presented.) curves homeomorphic to the unit circle, and assume that (Formula presented.). For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in (Formula presented.) and (Formula presented.), and furthermore that the space of (Formula presented.) differentials in (Formula presented.) decomposes as a direct sum of infinitesimally trivial differentials and (Formula presented.) harmonic (Formula presented.) differentials. Thus the tangent space of this Teichmüller space is given by (Formula presented.) harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of (Formula presented.) harmonic Beltrami differentials.
  • Dirichlet spaces of domains bounded by quasicircles
    (2019) Radnell, David; Schippers, Eric; Staubach, Wolfgang
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    Consider a multiply-connected domain ∑ in the sphere bounded by n non-intersecting quasicircles. We characterize the Dirichlet space of ∑ as an isomorphic image of a direct sum of Dirichlet spaces of the disk under a generalized Faber operator. This Faber operator is constructed using a jump formula for quasicircles and certain spaces of boundary values. Thereafter, we define a Grunsky operator on direct sums of Dirichlet spaces of the disk, and give a second characterization of the Dirichlet space of ∑ as the graph of the generalized Grunsky operator in direct sums of the space 1/2(1) on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.
  • A Model of the Teichmüller space of genus-zero bordered surfaces by period maps
    (2019-02-27) Radnell, David; Schippers, Eric; Staubach, Wolfgang
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We consider Riemann surfaces S with Σ borders homeomorphic to S 1 and no handles. Using generalized Grunsky operators, we define a period mapping from the infinite-dimensional Teichmüller space of surfaces of this type into the unit ball in the linear space of operators on an n-fold direct sum of Bergman spaces of the disk. We show that this period mapping is holomorphic and injective.
  • Quasiconformal maps of bordered Riemann surfaces with L2 Beltrami differentials
    (2017-06-29) Radnell, David; Schippers, Eric; Staubach, Wolfgang
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    Let Σ be a Riemann surface of genus g bordered by n curves homeomorphic to the circle S1. Consider quasiconformal maps f: Σ→Σ1 such that the restriction to each boundary curve is a Weil-Petersson class quasisymmetry. We show that any such f is homotopic to a quasiconformal map whose Beltrami differential is L2 with respect to the hyperbolic metric on Σ. The homotopy H(t, •): Σ → Σ1 is independent of t on the boundary curves; that is, H(t, p) = f(p) for all p ∈ ∂Σ.
  • Schiffer operators and calculation of a determinant line in conformal field theory
    (2021) Radnell, David; Schippers, Eric; Shirazi, Mohammad; Staubach, Wolfgang
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We consider an operator associated to compact Riemann surfaces endowed with a conformal map, f, from the unit disk into the surface, which arises in conformal field theory. This operator projects holomorphic functions on the surface minus the image of the conformal map onto the set of functions h so that the Fourier series h o f has only negative powers. We give an explicit characterization of the cokernel, kernel, and determinant line of this operator in terms of natural operators in function theory.
  • Slit-Strip Ising Boundary Conformal Field Theory 1: Discrete and Continuous Function Spaces
    (2022-12) Ameen, Taha; Kytölä, Kalle; Park, S. C.; Radnell, David
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of holomorphic functions in continuum domains as well as corresponding spaces of discrete holomorphic functions in lattice domains. We find distinguished sets of functions characterized by their singular behavior in the three infinite directions in the slit-strip domains, and note in particular that natural subsets of these functions span analogues of Hardy spaces. We prove convergence results of the distinguished discrete holomorphic functions to the continuum ones. In the subsequent articles, the discrete holomorphic functions will be used for the calculation of the Ising model fusion coefficients (as well as for the diagonalization of the Ising transfer matrix), and the convergence of the functions is used to prove the convergence of the fusion coefficients. It will also be shown that the vertex operator algebra of the boundary conformal field theory can be recovered from the limit of the fusion coefficients via geometric transformations involving the distinguished continuum functions.
  • Weil–Petersson class non-overlapping mappings into a Riemann surface
    (2016-08) Radnell, David; Schippers, Eric; Staubach, Wolfgang
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    For a compact Riemann surface of genus g with n punctures, consider the class of n-tuples of conformal mappings (φ1, . . . , φn) of the unit disk each taking 0 to a puncture. Assume further that (1) these maps are quasiconformally extendible to C , (2) the pre-Schwarzian of each φi is in the Bergman space, and (3) the images of the closures of the disk do not intersect. We show that the class of such non-overlapping mappings is a complex Hilbert manifold.