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Kotaro Yamada

Publications and source records attributed to Kotaro Yamada.

At least 37 records · Page 2Linked to original sources
arXiv2018

Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space

Catenoids in de Sitter $3$-space $S^3_1$ belong to a certain class of space-like constant mean curvature one surfaces. In a previous work, the authors classified such catenoids, and found that two different classes of countably many exceptional elliptic catenoids are not realized as closed subsets in $S^3_1$. Here we show that such exceptional catenoids have closed analytic extensions in $S^3_1$ with interesting properties.

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arXiv2017

Surfaces with light-like points in Lorentz-Minkowski 3-space with applications

With several concrete examples of zero mean curvature surfaces in $\boldsymbol{R}^3_1$ containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first fundamental form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in $\boldsymbol{R}^3_1$ whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.

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arXiv2017

Quadrics and Scherk towers

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net.

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arXiv2016

Zero mean curvature entire graphs of mixed type in Lorentz-Minkowski 3-space

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found two zero mean curvature entire graphs of mixed type that are not planes. As far as the authors know, these two examples were the only known examples of entire zero mean curvature graphs of mixed type without singularities. In this paper, we construct several families of real analytic zero mean curvature entire graphs of mixed type in Lorentz-Minkowski $3$-space. The entire graphs mentioned above lie in one of these classes.

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arXiv2016

Isometric realization of cross caps as formal power series and its applications

Two cross caps in Euclidean $3$-space are said to be formally isometric if their Taylor expansions of the first fundamental forms coincide by taking a suitable local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are formally isometric to $f$. As an application, we give a countable family of intrinsic invariants of cross caps which recognizes formal isometry classes completely.

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arXiv2015

Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $κ_Π$ called the product curvature (resp. $μ_Π$ called normalized product curvature) of $κ_c$ (resp. $μ_c$) and the limiting normal curvature $κ_ν$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of $κ_Π$ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.

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arXiv2015

Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2π/n)$-rotation symmetry with respect to its axis. In this paper, we show that the corresponding maximal surface $f_n$ in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ has an analytic extension $\tilde f_n$ as a properly embedded zero mean curvature surface. The extension changes type into a time-like (minimal) surface.

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arXiv2015

Mixed type surfaces with bounded mean curvature in 3-dimensional space-times

In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvature and investigate their properties.

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arXiv2015

An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications

In a previous work, the authors introduced the notion of `coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss-Bonnet formulas on coherent tangent bundles on 22-dimensional manifolds were proven, and several applications to surface theory were given. Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal E$ be a vector bundle of rank $n$ over $M^n$, and $φ:TM^n\to \mathcal E$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss-Bonnet formulas can be generalized to an index formula for the bundle homomorphism $φ$ under the assumption that $φ$ admits only certain kinds of generic singularities. We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.

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arXiv2014

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 $\boldsymbol{R}^3$ 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.

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arXiv2014

Isometric deformations of cuspidal edges

Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field $ν$ is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regular space curve) is not orthogonal to $ν$. This genericity is equivalent to the condition that its limiting normal curvature $κ_ν$ takes a non-zero value. In this paper, we show that a given generic (real analytic) cuspidal edge can be isometrically deformed preserving $κ_ν$ into a cuspidal edge whose singular set lies in a plane. Such a limiting cuspidal edge is uniquely determined from the initial germ of the cuspidal edge.

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arXiv2013

Zero mean curvature surfaces in Lorentz-Minkowski 3-space and 2-dimensional fluid mechanics

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon and its interesting connection to 2-dimensional fluid mechanics in this expository article.

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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.

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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.

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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.

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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.

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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).

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