Eric Schippers

Distortion theorems for higher order Schwarzian derivatives of univalent functions

Let S denote the class of functions which are univalent and holomorphic on the unit disc. We derive a simple differential equation for the Loewner flow of the Schwarzian derivative of a given f ∈ S. This is used to prove bounds on higher order Schwarzian derivatives which are sharp for the Koebe function. As well we prove some two-point distortion theorems for the higher order Schwarzians in terms of the hyperbolic metric.

Fiber structure and local coordinates for the Teichmüller space of a bordered Riemann surface

We show that the infinite-dimensional Teichmueller space of a Riemann surface whose boundary consists of n closed curves is a holomorphic fiber space over the Teichmueller space of n-punctured surfaces. Each fiber is a complex Banach manifold modeled on a two-dimensional extension of the universal Teichmueller space. The local model of the fiber, together with the coordinates from internal Schiffer variation, provides new holomorphic local coordinates for the infinite-dimensional Teichmueller space.

Conformal invariants and higher-order schwarz lemmas

We derive a generalization of the Grunsky inequalities using the Dirichlet principle. As a corollary, sharp distortion theorems for bounded univalent functions are proven for invariant differential expressions which are higher-order versions of the Schwarzian derivative. These distortion theorems can be written entirely in terms of conformai invariants depending on the derivatives of the hyperbolic metric, and can be interpreted as ’Schwarz lemmas’. In particular, sharp estimates on distortion of the derivatives of geodesic curvature of a curve under bounded univalent maps are given.

A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmuller space of such Riemann surfaces (which we refer to as the WP-class Teichmuller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmuller space is a Hilbert manifold. The inclusion map from the refined Teichmuller space into the usual Teichmuller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichmuller space by a properly discontinuous group of biholomorphisms.

Weil–Petersson class non-overlapping mappings into a Riemann surface

For a compact Riemann surface of genus g with n punctures, consider the class of n-tuples of conformal mappings (phi(1),..., phi(n)) of the unit disk each taking 0 to a puncture. Assume further tha ...

Behaviour of Kernel Functions under Homotopic Variations of Planar Domains

A variational formula is derived for Green’s function of multiply connected planar domains under homotopy of the boundary. The formula shows that up to first order, a homotopy behaves like the Hadamard variation. This is applied to show that certain expressions in the derivatives of Green’s function are monotonic with respect to set inclusion.

A symplectic functional analytic proof of the conformal welding theorem

We give a new functional-analytic/symplectic geometric proof of the conformal welding theorem. This is accomplished by representing composition by a quasisymmetric map phi as an operator on a suitable Hilbert space and algebraically solving the conformal welding equation for the unknown maps f and g satisfying g o phi = f. The univalence and quasiconformal extendibility of f and g is demonstrated through the use of the Grunsky matrix.

The Power Matrix, coadjoint action and quadratic differentials

The coefficients of a quadratic differential which is changing under the Loewner flow satisfy a well-known differential system studied by Schiffer, Schaeffer and Spencer, and others. By work of Roth, this differential system can be interpreted as Hamiltons equations. We apply the power matrix to interpret this differential system in terms of the coadjoint action of the matrix group on the dual of its Lie algebra. As an application, we derive a set of integral invariants of Hamiltons equations which is in a certain sense complete. In function theoretic terms, these are expressions in the coefficients of the quadratic differential and Loewner map which are independent of the parameter in the Loewner flow.

Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

Consider a Riemann surface of genus g bordered by n curves homeomorphic to the unit circle, and assume that 2g − 2 + n > 0. For such bordered Riemann surfaces, the authors have previously defined a Teichmuller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmuller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in L2 and L∞, and furthermore that the space of (−1, 1) differentials in L2 ∩ L∞ decomposes as a direct sum of infinitesimally trivial differentials and L2 harmonic (−1, 1) differentials. Thus the tangent space of this Teichmuller space is given by L2 harmonic Beltrami differentials. We conclude that this Teichmuller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmuller space is locally modeled on a space of L2 harmonic Beltrami differentials.

A Power Matrix Approach to the Witt Algebra and Loewner Equations

The theory of formal power series and derivations is developed from the point of view of the power matrix. A Loewner equation for formal power series is introduced. We then show that the matrix exponential is surjective onto the group of power matrices, and the coefficients are entire functions of finitely many coefficients of the infinitesimal generator. Furthermore coefficients of the solution to the Loewner equation with constant infinitesimal generator can be obtained by exponentiating an infinitesimal power matrix. We also use the formal Loewner equations to investigate the relation between holomorphicity of an infinitesimal generator to holomorphicity of the exponentiated matrix.