Searcharxiv⌕ Search

arXiv subjects

Masaaki Umehara

Publications and source records attributed to Masaaki Umehara.

At least 55 records · Page 3Linked to original sources
arXiv2013

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.

math.DG↗
arXiv2013

A construction of a complete bounded null curve in $\mathbb{C}^3$

We construct a complete bounded immersed null holomorphic curve in C^3, which is a recovery of the previous version of the paper Calc. Var. and PDE's vol 36 (2009); Erratum: to appear in Calc. Var. and PDE's, doi:10.1007/s00526-009-0226-5 by the last three authors on this subject.

math.DG↗
arXiv2012

Intrinsic Invariants of Cross Caps

It is classically known that generic smooth maps of R^2 into R^3 admit only cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap (u,uv,v^2) has non-trivial isometric deformations with infinite dimensional freedom. Since there are several geometric invariants for cross caps, the existence of isometric deformations suggests that one can ask which invariants of cross caps are intrinsic. In this paper, we show that there are three fundamental intrinsic invariants for cross caps. The existence of extrinsic invariants is also shown.

math.DG↗
arXiv2012

Calabi-Yau Problem for Legendrian curves in C^3 and applications

We construct a complete, bounded Legendrian immersion in C^3. As direct applications of it, we show the first examples of a weakly complete bounded flat front in hyperbolic 3-space, a weakly complete bounded flat front in de Sitter 3-space, and a weakly complete bounded improper affine front in R^3.

math.DG↗
arXiv2012

Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as an application of the minimal surface theory. This looks an interesting phenomenon if one comparing the fact that there are no complete minimal (resp. constant mean curvature one) surfaces in R^3 (resp. H^3) having bounded Gauss maps (resp. bounded hyperbolic Gauss maps).

math.DG↗
arXiv2011

CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere

CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 trinoids in H^3 that includes the reducible case.

math.DG↗
arXiv2011

Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces

We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via another hyperbolic Gauss map which is holomorphic. Using this, we show the orientability and co-orientability of CMC-1 faces (i.e., constant mean curvature one surfaces with admissible singular points) in de Sitter 3-space. (CMC-1 faces might not be wave fronts in general, but belong to a class of linear Weingarten surfaces with singular points.) Since both linear Weingarten fronts and CMC-1 faces may have singular points, orientability and co-orientability are both nontrivial properties. We also remark on some properties of non-orientable maximal surfaces in Lorentz-Minkowski 3-space, comparing the corresponding properties of CMC-1 faces in de Sitter 3-space.

math.DG↗
arXiv2011

Closed planar curves without inflections

We define a computable topological invariant $μ(γ)$ for generic closed planar regular curves $γ$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $μ(γ)$ on a given $γ$.

math.DG↗
arXiv2011

The behaviour of curvature functions at cusps and inflection points

At a 3/2-cusp of a given plane curve $γ(t)$, both of the Euclidean curvature $κ_g$ and the affine curvature $κ_A$ diverge. In this paper, we show that each of $\sqrt{|s_g|}κ_g$ and $(s_A)^2 κ_A$ (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a smooth function of the variable $t$, where $s_g$ (resp. $s_A$) is the Euclidean (resp. affine) arclength parameter of the curve corresponding to the 3/2-cusp $s_g=0$ (resp. $s_A=0$). Moreover, we give a characterization of the behaviour of the curvature functions $κ_g$ and $κ_A$ at 3/2-cusps. On the other hand, inflection points are also singular points of curves in affine geometry. We give a similar characterization of affine curvature functions near generic inflection points. As an application, new affine invariants of 3/2-cusps and generic inflection points are given.

math.DG↗
arXiv2010

Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts

We give a definition of `coherent tangent bundles', which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2-manifold, there are two Gauss-Bonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional Gauss-Bonnet formulas for the third fundamental form. Surprisingly, these are different from those for the first fundamental form, and using these four formulas, we get several new results on the topology and geometry of wave fronts.

math.DG↗
arXiv2010

A_2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space

In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A_2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space.

math.DG↗
arXiv2010

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 puntured disk.

math.DG↗
arXiv2010

The duality of conformally flat manifolds

In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points (i.e. admissible GCF-manifolds), * frontals as hypersurfaces in the lightcone Q^{n+1}_+, * frontals as hypersurfaces in the hyperbolic space H^{n+1}, * sapacelike frontals as hypersurfaces in the de Sitter space S^{n+1}_1. Recently, the duality of conformally flat Riemannian manifolds was found by several geometers. In our setting, this duality can be explained via the existence of a two-fold map of the congruent classes of admissible GCF-manifolds into that of frontals in H^{n+1}. It should be remarked that the dual conformally flat metric may have degenerate points even when the original conformally flat metric is positive definite. This is the reason why we consider conformally flat manifolds with singular points. In fact, the duality is an involution on the set of admissible GCF-manifolds. The case n=2 requires an exceptional treatment, since any Riemannian 2-manifolds are conformally flat. At the end of this paper, we also determine the moduli space of isometric immersions of a given simply connected Riemannian 2-manifold into the lightcone Q^{3}_+.

math.DG↗
arXiv2010

The duality between singular points and inflection points on wave fronts

In the previous paper (arXiv:0804.0701), the authors gave criteria for A_{k+1}-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 M^2 is a compact orientable 2-manifold, and M_-$is the open subset of M^2 where the Hessian of f takes negative values. This is a generalization of Bleecker and Wilson's formula for immersed surfaces in the affine 3-space.

math.DG↗
arXiv2010

Singularities of Blaschke normal maps of convex surfaces

We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.

math.DG↗
arXiv2009

New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter 3-space

Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one surfaces with admissible singularities in de Sitter 3-space) and obtain "cousins" of those maximal surfaces. Cone-like singular points on maximal surfaces are very important, although they are not stable under perturbations of maximal surfaces. It is interesting to ask if cone-like singular points can appear on a maximal surface having other kinds of singularities. Until now, no such examples were known. We also construct a family of complete maximal surfaces with two complete ends and with both cone-like singular points and cuspidal edges.

math.DG↗
arXiv2009

Spacelike mean curvature one surfaces in de Sitter 3-space

The first author studied spacelike constant mean curvature one (CMC-1) surfaces in de Sitter 3-space when the surfaces have no singularities except within some compact subset and are of finite total curvature on the complement of this compact subset. However, there are many CMC-1 surfaces whose singular sets are not compact. In fact, such examples have already appeared in the construction of trinoids given by Lee and the last author via hypergeometric functions. In this paper, we improve the Osserman-type inequality given by the first author. Moreover, we shall develop a fundamental framework that allows the singular set to be non-compact, and then will use it to investigate the global behavior of CMC-1 surfaces.

math.DG↗