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Masaaki Umehara

Publications and source records attributed to Masaaki Umehara.

At least 73 records · Page 4Linked to original sources
arXiv2009

Asymptotic behavior of flat surfaces in hyperbolic 3-space

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called "pitch" p) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have -1<p<=0. If the singular set accumulates at the end, the pitch p is a positive rational number not equal to 1. Choosing appropriate positive integers n and m so that p=n/m, suitable slices of the end by horospheres are asymptotic to d-coverings (d-times wrapped coverings) of epicycloids or d-coverings of hypocycloids with 2n_0 cusps and whose normal directions have winding number m_0, where n=n_0d, m=m_0d (n_0, m_0 are integers or half-integers) and d is the greatest common divisor of m-n and m+n. Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts.

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arXiv2008

Flat Surfaces with singularities in Euclidean 3-space

It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $ν$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,ν)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fonts, and give a representation formula for complete flat fronts. As an application, we show that a complete flat front has properly embedded ends if and only if its Gauss image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.

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arXiv2008

Complete bounded holomorphic curves immersed in C^2 with arbitrary genus

In the previous paper, the authors constructed a complete holomorphic immersion of the unit disk D into C^2 whose image is bounded. In this paper, we shall prove existence of complete 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 used by F. J. Lopez to perturb the genus zero example changing its genus. As an analogue the above construction, we also give a new method to construct complete bounded minimal immersions (resp. weakly complete maximal surface) with arbitrary genus and finite topology in Euclidean 3-space (resp. Lorentz-Minkowski 3-spacetime).

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arXiv2008

A_k singularities of wave fronts

In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the geometry of fronts" for surfaces. The crucial point to prove our criteria for A_k-singularities is to introduce a suitable parametrization of the singularities called the "k-th KRSUY-coordinates". Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a real (resp. complex) hypersurface (as a wave front) in R^{n+1} (resp. C^{n+1}) is differentiably (resp. holomorphically) right-left equivalent to the A_{k+1}-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane R^n (resp. C^n) is right-left equivalent to the A_k-type singular point in R^n (resp. C^{n}). Moreover, we show that the restriction of a C-infinity-map f:R^n --> R^n to its Morin singular set gives a wave front consisting of only A_k-type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.

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arXiv2008

Constructing mean curvature 1 surfaces in $H^3$ with irregular ends

With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an essential singularity, resp. at most a pole, there.) There are some known surfaces with irregular ends, but they are all either reducible or of infinite total curvature. (The surface is reducible if and only if the monodromy of the secondary Gauss map can be simultaneously diagonalized.) Up to now there have been no known complete irreducible CMC 1 surfaces in $H^3$ with finite total curvature and irregular ends. The purpose of this paper is to construct countably many 1-parameter families of genus zero CMC 1 surfaces with irregular ends and finite total curvature, which have either dihedral or Platonic symmetries. For all the examples we produce, we show that they have finite total curvature and irregular ends. For the examples with dihedral symmetry and the simplest example with tetrahedral symmetry, we show irreducibility. Moreover, we construct a genus one CMC 1 surface with four irregular ends, which is the first known example with positive genus whose ends are all irregular.

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arXiv2008

Period problems for mean curvature one surfaces in $H^3$ (with application to surfaces of low total curvature)

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature, which is the total absolute curvature of the dual CMC 1 surface. Here we discuss results on both notions (proven in two other papers by the authors), and we introduce new results as well.

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arXiv2007

Singularities of Maximal Surfaces

We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.

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arXiv2007

Behavior of corank one singular points on wave fronts

Let $M^2$ be an oriented 2-manifold and $f:M^2\to R^3$ a $C^\infty$-map. A point $p\in M^2$ is called a singular point if $f$ is not an immersion at $p$. The map $f$ is called a front (or wave front), if there exists a unit $C^\infty$-vector field $ν$ such that the image of each tangent vector $df(X)$ $(X\in TM^2)$ is perpendicular to $ν$, and the pair $(f,ν)$ gives an immersion into $R^3\times S^2$. In our previous paper, we gave an intrinsic formulation of wave fronts in $R^3$. In this paper, we shall investigate the behavior of cuspidal edges near corank one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation.

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arXiv2006

The geometry of fronts

We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails.

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arXiv2006

Inflection points and double tangents on anti-convex curves in the real projective plane

A simple closed curve $γ$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(γ)-2δ(γ)=3$ for anti-convex curves, where $i(γ)$ is the number of independent (true) inflection points and $δ(γ)$ the number of independent double tangents. This formula is a refinement of the classical Möbius theorem. We shall also show that there are three inflection points on a given anti-convex curve such that the tangent lines at these three inflection points cross the curve only once. Our approach is axiomatic and can be applied in other situations. For example, we prove similar results for curves of constant width as a corollary.

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arXiv2006

Flat fronts in hyperbolic 3-space and their caustics

After Galvez, Martinez and Milan discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space, the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H^3. In the present work we broaden the notion of completeness to weak completeness, and of front to p-front. As a front is a p-front and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces. This more general class contains the caustics of flat fronts -- shown also to be flat by Roitman (who gave a holomorphic representation formula for them) -- which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only p-fronts. Using the new framework, we obtain characterizations for caustics.

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arXiv2005

Maximal surfaces with singularities in Minkowski space

We shall investigate maximal surfaces in Minkowski 3-space with singularities. Although the plane is the only complete maximal surface without singular points, there are many other complete maximal surfaces with singularities and we show that they satisfy an Osserman-type inequality.

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arXiv2004

Singularities of flat fronts in hyperbolic 3-space

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a (wave) front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We shall give easily-computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we shall prove that generically flat fronts in the hyperbolic 3-space admit only cuspidal edges and swallowtails. Moreover, we will show that every complete flat front (which is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

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arXiv2003

Flat fronts in hyperbolic 3-space

We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded. Moreover, we shall give new examples for which equality holds.

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arXiv2003

An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)

For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementary proof of Small's formula. Moreover, a similar formula for Legendrian curves in PSL(2,C) is given. As null curves in PSL(2,C) are related to mean curvature one surfaces in hyperbolic 3-space H^3, Legendrian curves are related to flat surfaces in H^3. So, as an application of Small-type formula for Lengendrian curves, we give new examples of flat surfaces in H^3.

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arXiv2001

An analogue of minimal surface theory in SL(n,C)/SU(n)

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semisimple Lie groups (e.g. SL(n,C)/SU(n)), which contains minimal surfaces in R^n and constant mean curvature 1 surfaces in H^3. A Weierstrass type representation formula, and a Chern-Osserman type inequality for such surfaces are given.

math.DG↗
arXiv2001

A global theory of flexes of periodic functions

For a real valued periodic smooth function u on R, $n\ge 0$, one defines the osculating polynomial $ϕ_s$ (of order 2n+1) at a point $s\in R$ to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on $S^1$ if and only if $ϕ_s\ge u$ (resp. $ϕ_s\le u$) and the preimage $(ϕ-u)^{-1}(0)$ is connected. We prove that any smooth periodic function u has at least n+1 clean maximal flexes of order 2n+1 and at least n+1 clean minimal flexes of order 2n+1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.

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