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 Teichmüller 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 Teichmüller space and as functions on the rigged moduli space of Segal and Vafa. The construction uses an identification of Teichmüller 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.

To a conformal map f from the disk 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(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(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.

In this paper, we consider a compact Riemann surface & Rscr; 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 L2 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.

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 Teichmüller space.

Consider a compact surface ℛ with distinguished points z_1,… ,z_n and conformal maps f_k from the unit disk into non-overlapping quasidisks on ℛ taking 0 to z_k . Let Σ be the Riemann surface obtained by removing the closures of the images of f_k from ℛ . We define forms which are meromorphic on ℛ with poles only at z_1,… ,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 Σ is uniquely expressible as a series of Faber–Tietz forms. This series converges both in L^2(Σ ) and uniformly on compact subsets of Σ .

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.

This is the second in a series of four papers developing a scattering theory for harmonic one-forms on Riemann surfaces. In this paper we develop a conformally invariant characterization of the Sobolev space H-1 / 2 (Gamma) where Gamma is a border of a Riemann surface which is homeomorphic to the circle. We show that the boundary values of L-2 harmonic one-forms are in H-1 / 2 (Gamma) . Also, let Sigma be a Riemann surface with a finite number of borders homeomorphic to the circle. We show that the Dirichlet problem on a Riemann surface Sigma with border) partial derivative Sigma for one-forms with boundary values in H-1 / 2 (partial derivative Sigma) and suitable cohomological data is well-posed. Furthermore, we prove the following "overfare" result. Let R be a compact Riemann surface split into two surfaces Sigma(1) and Sigma(2) by a complex of quasicircles. Given an L-2 harmonic one-form alpha(1) on Sigma(1) , there is a unique L-2 harmonic one-form alpha(2) on Sigma(2) with the same boundary values in the above sense.

This paper gives an overview of our work on a scattering theory of one-forms and functions in a system of quasicircles on Riemann surfaces. It is rooted in an “overfare” process which takes a harmonic function on one side of the system of quasicircles to a harmonic function on the other side, with the same boundary values in a certain intrinsic non-tangential sense. This is bounded with respect to Dirichlet energy. If extra cohomological data is specified, one can apply this process to harmonic one-forms, and the resulting “scattering matrix” in terms of the holomorphic and anti-holomorphic components of the one-form is unitary. We describe applications to approximation theory, global analysis of singular integral operators on Riemann surfaces, and a new extension of the classical period map to surfaces of genus g with n boundary curves.

Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichmüller spaces of infinite type, including for example the universal Teichmüller space. In this paper we give a survey of the main results in the Weil-Petersson geometry of infinite-dimensional Teichmüller spaces. This includes the rigorous definition of complex Hilbert manifold structures, Kähler 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 Teichmüller space in other fields such as fluid mechanics and two-dimensional conformal field theory are also provided.

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.

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.

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.

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 $\Sigma$, 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 $\Sigma$ 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 $\Sigma$ by elements of Bergman space and Dirichlet space on fixed regions in $R$ containing $\Sigma$.

Consider a multiply-connected domain [Formula: see text] in the sphere bounded by [Formula: see text] non-intersecting quasicircles. We characterize the Dirichlet space of [Formula: see text] 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 [Formula: see text] as the graph of the generalized Grunsky operator in direct sums of the space [Formula: see text] on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

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.

Let $R$ be a compact Riemann surface and $\Gamma$ be a Jordan curve separating $R$ into connected components $\Sigma_1$ and $\Sigma_2$. We consider Calder\'on-Zygmund type operators $T(\Sigma_1,\Sigma_k)$ taking the space of $L^2$ anti-holomorphic one-forms on $\Sigma_1$ to the space of $L^2$ holomorphic one-forms on $\Sigma_k$, which we call the Schiffer operators. We extend results of Menahem M. Schiffer and others, which where confined to analytic Jordan curves $\Gamma$, 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 $\Sigma_1$, then the Schiffer operator $T(\Sigma_1,\Sigma_2)$ is an isomorphism onto the set of exact one-forms on $\Sigma_2$. 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.

We consider Riemann surfaces Σ \Sigma with n n borders homeomorphic to S 1 \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 n -fold direct sum of Bergman spaces of the disk. We show that this period mapping is holomorphic and injective.

Let \(\Gamma \) be a bounded Jordan curve with complementary components \(\Omega ^{\pm }\). We show that the jump decomposition is an isomorphism if and only if \(\Gamma \) is a quasicircle. We also show that the Bergman space of \(L^{2}\) harmonic one-forms on \(\Omega ^{+}\) is isomorphic to the direct sum of the holomorphic Bergman spaces on \(\Omega ^{+}\) and \(\Omega ^{-}\) if and only if \(\Gamma \) is a quasicircle. This allows us to derive various relations between a reflection of harmonic functions in quasicircles and the jump decomposition on the one hand, and the Grunsky operator, Faber series and kernel functions of Schiffer on the other hand. It also leads to new interpretations of the Grunsky and Schiffer operators. We show throughout that the most general setting for these relations is quasidisks.

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.

We associate a functional of pairs of simply-connected regions D 2 ⊆ D 1 to any quadratic differential on D 1 with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form ( ∂q ) 2 for a singular harmonic function q on D 1 . 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 [16].