David Radnell

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.

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 ...

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.

The semigroup of rigged annuli and the Teichmüller space of the annulus

Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show that this semigroup is a quotient of the Teichmueller space of doubly-connected Riemann surfaces by a Z action. Furthermore, the semigroup can be given a complex structure in two distinct, natural ways. We show that these two complex structures are equivalent, and furthermore that multiplication is holomorphic. Finally, we show that the class of quasiconformally-extendible conformal maps of the disk to itself is a complex submanifold in which composition is holomorphic.