Ege Fujikawa

Smale’s mean value conjecture and the coefficients of univalent functions

We study Smales mean value conjecture and its connection with the second coefficients of univalent functions. We improve the bound on Smales constant given by Beardon, Minda and Ng.

Limit sets and regions of discontinuity of Teichmuller modular groups

For a Riemann surface of infinite type, the Teichmiiller modular group does not act properly discontinuously on the Teichmuller space, in general. As an analogy to the theory of Kleinian groups, we divide the Teichmuller space into the limit set and the region of discontinuity for the Teichmiiller modular group, and observe their properties.

Pure mapping class group acting on Teichmüller space

For a Riemann surface of analytically infinite type, the action of the quasiconformal mapping class group on the Teichmuller space is not discontinuous in general. In this paper, we consider pure mapping classes that fix all topological ends of a Riemann surface and prove that the pure mapping class group acts on the Teichmuller space discontinuously under a certain geometric condition of a Riemann surface. We also consider the action of the quasiconformal mapping class group on the asymptotic Teichmuller space. Non-trivial mapping classes can act on the asymptotic Teichmuller space trivially. We prove that all such mapping classes are contained in the pure mapping class group.

The Nielsen realization problem for asymptotic Teichmüller modular groups

Under a certain geometric assumption on a hyperbolic Riemann surface, we prove an asymptotic version of the fixed point theorem for the Teichmüller modular group, which asserts that every finite subgroup of the asymptotic Teichmüller modular group has a common fixed point in the asymptotic Teichmüller space. For its proof, we use a topological characterization of the asymptotically trivial mapping class group, which has been obtained in the authors’ previous paper, but a simpler argument is given here. As a consequence, every finite subgroup of the asymptotic Teichmüller modular group is realized as a group of quasiconformal automorphisms modulo coincidence near infinity. Furthermore, every finite subgroup of a certain geometric automorphism group of the asymptotic Teichmüller space is realized as an automorphism group of the Royden boundary of the Riemann surface. These results can be regarded as asymptotic versions of the Nielsen realization theorem.

Discreteness of the Mapping Class Group for Riemann Surfaces of Infinite Analytic Type

In the theory of Teichmuller spaces of compact Riemann surfaces, the mapping class group (or modular group) plays an important role. Indeed, the group is just equal to the group of biholomorphic automorphisms of the Teichmuller space and it acts discontinuously(cf. [3], [4]).