Value distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space

We give an effective estimate for the totally ramified value number of the hyperbolic Gauss maps of complete flat fronts in the hyperbolic three-space. As a corollary, we give the upper bound of the number of exceptional values of them for some topological cases. Moreover, we obtain some new examples for this class.Comment: 14 pages, to appear in Houston Journal of Mathematic

Anyon Basis of c=1 Conformal Field Theory

We study the $c=1$ conformal field theory of a free compactified boson with radius $r=\sqrt{\beta}$ ($\beta$ is an integer). The Fock space of this boson is constructed in terms of anyon vertex operators and each state is labeled by an infinite set of pseudo-momenta of filled particles in pseudo-Dirac sea. Wave function of multi anyon state is described by an eigenfunction of the Calogero-Sutherland (CS) model. The $c=1$ conformal field theory at $r=\sqrt{\beta}$ gives a field theory of CS model. This is a natural generalization of the boson-fermion correspondence in one dimension to boson-anyon correspondence. There is also an interesting duality between anyon with statistics $\theta=\pi/\beta$ and particle with statistics $\theta=\beta \pi$.Comment: 17 page

Function-theoretic properties for the Gauss maps of various classes of surfaces

We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1205.478

Ramification estimates for the hyperbolic Gauss map

We give the best possible upper bound on the number of exceptional values and the totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic constant mean curvature one surfaces in the hyperbolic three-space and some partial results on the Osserman problem for algebraic case. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.Comment: 16 pages, corrected some typos. OCAMI Preprint Series 08-1, to appear in Osaka Journal of Mathematic

Four-Dimensional Painlev\'e-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems

This is the last part of a series of three papers entitled "Four-dimensional Painlev\'e-type equations associated with ramified linear equations". In this series of papers we aim to construct the complete degeneration scheme of four-dimensional Painlev\'e-type equations. In the present paper, we consider the degeneration of the Garnier system in two variables and the Fuji-Suzuki system

On the maximal number of exceptional values of Gauss maps for various classes of surfaces

The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and weakly complete flat surfaces in the hyperbolic three-space. For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface.Comment: 13 pages, to appear in Mathematische Zeitschrif