Masaaki Umehara

Complete bounded holomorphic curves immersed in C2 with arbitrary genus

In (MUY), a complete holomorphic immersion of the unit disk D into C 2 whose image is bounded was constructed. In this paper, we shall prove existence of com- plete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology, whose image is bounded in C 2 . To construct such immersions, we apply the method in (L) to perturb the genus zero example in (MUY) changing its genus. As an analogue the above construction, we also give a new method to construct com- plete bounded minimal immersions (resp. weakly complete maximal surface) with arbi- trary genus and finite topology in Euclidean 3-space (resp. L orentz-Minkowski 3-space- time).

Applications of a completeness lemma in minimal surface theory to various classes of surfaces

We give several applications of a lemma on completeness used by Osserman to show the meromorphicity of Weierstrass data for complete minimal surfaces with finite total curvature. Completeness and weak completeness are defined for several classes of surfaces which admit singular points. The completeness lemma is a useful machinery for the study of completeness in these classes of surfaces. In particular, we show that a constant mean curvature one (i.e. CMC-1) surface in de Sitter 3-space is complete if and only if it is weakly complete, the singular set is compact and all the ends are conformally equivalent to a punctured disk.

The duality between singular points and inflection points on wave fronts

In the previous paper, the authors gave criteria for AkC1-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of 2-inflection points (i.e. godron points) on an immersed surface in the real projective space is equal to the Euler number of M . Here M2 is a compact orientable 2-manifold, and M is the open subset of M2 where the Hessian of f takes negative values. This is a generalization of Bleecker and Wilson’s formula [3] for immersed surfaces in the affine 3-space.

Zero mean curvature surfaces in Lorentz--Minkowski 3-space which change type across a light-like line

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been analyzed before. As a continuation of a previous work by the authors, we give the first example of a family of such surfaces which change type across the light-like line. As a corollary, we also obtain a family of zero mean curvature hypersurfaces in R^{n+1}_1 that change type across an (n-1)-dimensional light-like plane.

A characterization of compact surfaces with constant mean curvature

Surfaces in a 3-space of constant curvature, whose arbitrary sufficiently small open subsets admit a non-trivial isometric deformation preserving the mean curvature function, are called locally H-deformable. It is well known that surfaces with constant mean curvature which are not totally umbilical are all locally H-deformable. Conversely, we shall show in this paper that any compact locally H-deformable surface has constant mean curvature.

Intrinsic properties of surfaces with singularities

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss–Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds.

Sextactic points on a simple closed curve

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

A deformation of tori with constant mean curvature in ³ to those in other space forms

It is shown that tori with constant mean curvature in R 3 constructed by Wente [7] can be deformed to tori with constant mean curvature in the hyperbolic 3-space or the 3-sphere

Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts

We define cuspidal curvature

An analogue of minimal surface theory in (

\kappa_c