arXiv subjects

Eric Schippers

Publications and source records attributed to Eric Schippers.

At least 19 recordsLinked to original sources

arXiv2026

A unified approach to conformal and modular invariants

In this paper we give a general family of conformal invariants associated to bordered Riemann surfaces endowed with boundary parametrizations, or equivalently compact surfaces endowed with conformal maps. Each invariant is specified by a field of one-forms over a Teichm\"uller space of infinite conformal type. The invariants are positive, and under certain conditions monotonic. It is shown that these conformal invariants can be viewed as generalized modular invariants on Teichm\"uller space and as functions on the rigged moduli space of Segal and Vafa. The construction uses an identification of Teichm\"uller space and the rigged moduli space, as well as analytic work of the authors showing that the transfer or ``overfare'' of harmonic functions sharing boundary values on a quasicircle is bounded. Demanding invariance under various subgroups of the modular group -- equivalently, under the group of quasisymmetric reparametrizations of a sub-collection of borders -- generates conformal invariants. We show that a wide variety of conformal invariants can be obtained through various choices of the field of one-forms. These include modules of doubly-connected domains, period mappings obtained from harmonic measures, inequalities for higher-order conformal invariants, and the Grunsky inequalities and their recent generalizations to Riemann surfaces.

math.DG↗

arXiv2025

Scattering theory on Riemann surfaces I: Schiffer operators, cohomology, and index theorems

We consider a compact Riemann surface $\mathscr{R}$ with a complex of non-intersecting Jordan curves, whose complement is a pair of Riemann surfaces with boundary, each of which may be possibly disconnected. We investigate conformally invariant integral operators of Schiffer, which act on $L^{2}$ anti-holomorphic one-forms on one of these surfaces with boundary and produce holomorphic one-forms on the disjoint union. These operators arise in potential theory, boundary value problems, approximation theory, and conformal field theory, and are closely related to a kind of Cauchy operator. We develop an extensive calculus for the Schiffer and Cauchy operators, including a number of adjoint identities for the Schiffer operators. In the case that the Jordan curves are quasicircles, we derive a Plemelj-Sokhotski jump formula for Dirichlet-bounded functions. We generalize a theorem of Napalkov and Yulmukhametov, which shows that a certain Schiffer operator is an isomorphism for quasicircles. Finally, we characterize the kernels and images, and derive index theorems for the Schiffer operators, which will in turn connect conformal invariants to topological invariants.

math.DG↗

arXiv2025

Scattering theory on Riemann surfaces II: The scattering matrix and generalized period mappings

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems involving systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of integral operators which we call Schiffer operators, and show that the matrix is unitary. We also obtain a general association of positive polarizing Lagrangian spaces to bordered Riemann surfaces, which unifies the classical polarizations for compact surfaces of algebraic geometry with the infinite-dimensional period map of the universal Teichm\"uller space.

math.DG↗

arXiv2023

A Fermionic Grunsky operator

To a conformal map $f$ from the disk $\mathbb{D}$ into the complex plane onto a domain with rectifiable Ahlfors-regular boundary, we associate a new kind of Grunsky operator on the Hardy space of the unit disk. This is analogous to the classical Grunsky operator, which itself can be viewed as an operator on Bergman or Dirichlet space. We show that the pull-back of the Smirnov space of the complement of $f(\mathbb{D})$ by $f$ is the graph of the Grunsky operator. We also characterize those domains with rectifiable Ahlfors-regular boundaries such that the Grunsky operator is Hilbert-Schmidt. In particular, we show that if the Grunsky operator is Hilbert-Schmidt, then $f(\mathbb{D})$ is a Weil-Petersson quasidisk. The formulations of the results and proofs make essential use of a geometric treatment of Smirnov space as a space of half-order differentials.

math.CV↗

arXiv2023

Grassmannians of Lagrangian Polarizations

This paper is an introduction to polarizations in the symplectic and orthogonal settings. They arise in association to a triple of compatible structures on a real vector space, consisting of an inner product, a symplectic form, and a complex structure. A polarization is a decomposition of the complexified vector space into the eigenspaces of the complex structure; this information is equivalent to the specification of a compatible triple. When either a symplectic form or inner product is fixed, one obtains a Grassmannian of polarizations. We give an exposition of this circle of ideas, emphasizing the symmetry of the symplectic and orthogonal settings, and allowing the possibility that the underlying vector spaces are infinite-dimensional. This introduction would be useful for those interested in applications of polarizations to representation theory, loop groups, complex geometry, moduli spaces, quantization, and conformal field theory.

math.DG↗

arXiv2023

Faber series for $L^2$ holomorphic one-forms on Riemann surfaces with boundary

Consider a compact surface $\mathscr{R}$ with distinguished points $z_1,\ldots,z_n$ and conformal maps $f_k$ from the unit disk into non-overlapping quasidisks on $\mathscr{R}$ taking $0$ to $z_k$. Let $\Sigma$ be the Riemann surface obtained by removing the closures of the images of $f_k$ from $\mathscr{R}$. We define forms which are meromorphic on $\mathscr{R}$ with poles only at $z_1,\ldots,z_n$, which we call Faber-Tietz forms. These are analogous to Faber polynomials in the sphere. We show that any $L^2$ holomorphic one-form on $\Sigma$ is uniquely expressible as a series of Faber-Tietz forms. This series converges both in $L^2(\Sigma)$ and uniformly on compact subsets of $\Sigma$.

math.CV↗

arXiv2023

Weil-Petersson Teichm\"uller theory of surfaces of infinite conformal type

Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichm\"uller spaces of infinite type, including for example the universal Teichm\"uller space. In this paper we give a survey of the main results in the Weil-Petersson geometry of infinite-dimensional Teichm\"uller spaces. This includes the rigorous definition of complex Hilbert manifold structures, K\"ahler geometry and global analysis, and generalizations of the period mapping. We also discuss the motivations of the theory in representation theory and physics beginning in the 1980s. Some examples of the appearance of Weil-Petersson Teichm\"uller space in other fields such as fluid mechanics and two-dimensional conformal field theory are also provided.

math.CV↗

arXiv2021

A scattering theory of harmonic one-forms on Riemann surfaces

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems through systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of integral operators which we call Schiffer operators, and show that the matrix is unitary. As a consequence of this scattering theory, we prove index theorems relating these conformally invariant integral operators to topological invariants. We also obtain a general association of positive polarizing Lagrangian spaces to bordered Riemann surfaces, which unifies the classical polarizations for compact surfaces of algebraic geometry with the infinite-dimensional period map of the universal Teichmueller space.

math.DG↗

arXiv2020

Analysis on quasidisks; a unified approach through transmission and jump problems

We give an exposition of results from a crossroad between geometric function theory, harmonic analysis, boundary value problems and approximation theory, which characterize quasicircles. We will specifically expose the interplay between the jump decomposition, singular integral operators and approximation by Faber series. Our unified point of view is made possible by the the concept of transmission.

math.CV↗

arXiv2020

Transmission of harmonic functions through quasicircles on compact Riemann surfaces

Let $R$ be a compact surface and let $Γ$ be a Jordan curve which separates $R$ into two connected components $Σ_1$ and $Σ_2$. A harmonic function $h_1$ on $Σ_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $Γ$. We show that if $Γ$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $Σ_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $Σ_1$ into $Σ_2$ is bounded with respect to the Dirichlet semi-norm.

math.CV↗

arXiv2019

Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $Σ$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $\mathcal{O}$ to the Bergman space of holomorphic forms on $Σ$ is an isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $Σ$ by elements of Bergman space and Dirichlet space on fixed regions in $R$ containing $Σ$.

math.CV↗

arXiv2018

Plemelj-Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operator

Let $R$ be a compact Riemann surface and $Γ$ be a Jordan curve separating $R$ into connected components $Σ_1$ and $Σ_2$. We consider Calderón-Zygmund type operators $T(Σ_1,Σ_k)$ taking the space of $L^2$ anti-holomorphic one-forms on $Σ_1$ to the space of $L^2$ holomorphic one-forms on $Σ_k$, which we call the Schiffer operators. We extend results of Menahem M. Schiffer and others, which where confined to analytic Jordan curves $Γ$, to general quasicircles in a characterizing manner, and prove new identities for adjoints of the Schiffer operators. Furthermore, we show that if $V$ is the space of anti-holomorphic one-forms orthogonal to $L^2$ forms on $R$ with respect to the inner product on $Σ_1$, then the Schiffer operator $T(Σ_1,Σ_2)$ is an isomorphism onto the set of exact one-forms on $Σ_2$. Using the relation between the Schiffer operator and a Cauchy-type integral involving Green's function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

math.CV↗

arXiv2018

Dirichlet space of domains bounded by quasicircles

Consider a multiply-connected domain $Σ$ in the sphere bounded by $n$ non-intersecting quasicircles. We characterize the Dirichlet space of $Σ$ 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 $Σ$ as the graph of the generalized Grunsky operator in direct sums of the space $\mathcal{H}^{1/2}(\mathbb{S}^1)$ on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

math.CV↗

arXiv2017

A Model of the Teichmüller space of genus-zero bordered surfaces by period maps

We consider Riemann surfaces $Σ$ with $n$ borders homeomorphic to $\mathbb{S}^1$ 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.

math.CV↗

arXiv2017

Comparison moduli spaces of Riemann surfaces

We define a kind of moduli space of nested surfaces and mappings, which we call a comparison moduli space. We review examples of such spaces in geometric function theory and modern Teichmueller theory, and illustrate how a wide range of phenomena in complex analysis are captured by this notion of moduli space. The paper includes a list of open problems in classical and modern function theory and Teichmueller theory ranging from general theoretical questions to specific technical problems.

math.CV↗

arXiv2017

Quasiconformal Teichmuller theory as an analytical foundation for two-dimensional conformal field theory

The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal Teichmuller theory and geometric function theory. In particular we propose that the natural analytic setting for conformal field theory is the moduli space of Riemann surfaces with so-called Weil-Petersson class parametrizations. A collection of rigorous analytic results is advanced here as evidence. This class of parametrizations has the required regularity for CFT on one hand, and on the other hand are natural and of interest in their own right in geometric function theory.

math.CV↗

arXiv2016

Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks

A complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values in a certain conformally invariant sense, by a construction of H. Osborn. We call the set of such boundary values the Douglas-Osborn space. One may then attempt to solve the Dirichlet problem on the complement for these boundary values. This defines a reflection of harmonic functions. We show that quasicircles are precisely those Jordan curves for which this reflection is defined and bounded. We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj-Sokhotski jump formula holds on quasicircles with boundary data in the Douglas-Osborn space. This enables us to prove the well-posedness of a Riemann-Hilbert problem with boundary data in the Douglas-Osborn space on quasicircles.

math.CV↗

arXiv2016

Conformal invariants associated with quadratic differentials

Z. Nehari developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle. Given a harmonic function with singularity on a domain $R$, it associates a monotonic functional of subdomains $D \subseteq R$. In the case that $R$ is conformally equivalent to a disk, we extend Nehari's method by associating a functional to any quadratic differential on $R$ with specified singularities. Nehari's method corresponds to the special case that the quadratic differential is of the form $(\partial q)^2$ for a singular harmonic function $q$ on $R$. Besides being more general, our formulation is conformally invariant, and has a particularly elegant equality statement. As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda.

math.CV↗