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The Gauss map of pseudo-algebraic minimal surfaces

2008-01-01
Yu Kawakami , Ryoichi Kobayashi , Reiko Miyaoka
We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally … We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a new proof of Fujimoto's theorem for this class, which not only simplifies the proof but also reveals the geometric meaning behind it.
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Entire zero-mean curvature graphs of mixed type in Lorentz–Minkowski 3-space

2016-10-18
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more

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

2011-08-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
CMC-1 trinoids (i.e. constant mean curvature one immersed surfaces of genus zero with three regular embedded ends) in hyperbolic 3-space $H^{3}$ are irreducible generically, and the irreducible ones have been … CMC-1 trinoids (i.e. constant mean curvature one immersed surfaces of genus zero 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 here we give an explicit description of CMC-1 trinoids in $H^{3}$ that includes the reducible case.
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On the totally ramified value number of the Gauss map of minimal surfaces

2006-01-01
Yu Kawakami
In this paper, we define the totally ramified value number (TRVN) of the Gauss map of a complete minimal surface, and prove that there exist algebraic minimal surfaces that have … In this paper, we define the totally ramified value number (TRVN) of the Gauss map of a complete minimal surface, and prove that there exist algebraic minimal surfaces that have the TRVN equal to $2.5$.
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On the maximal number of exceptional values of Gauss maps for various classes of surfaces

2012-12-06
Yu Kawakami
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The Gauss map of pseudo‐algebraic minimal surfaces in R<sup>4</sup>

2009-01-20
Yu Kawakami
Abstract In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number of the Gauss map for pseudo‐algebraic and algebraic minimal surfaces … Abstract In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number of the Gauss map for pseudo‐algebraic and algebraic minimal surfaces in Euclidean four‐space and give a kind of unicity theorem (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

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

2017-04-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 $(2\pi/n)$-rotation symmetry with respect to its … 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 $(2\pi/n)$-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.

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

2015-02-14
Yu Kawakami
Abstract 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, … Abstract 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.
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Value distribution of the Gauss map of improper affine spheres

2012-07-01
Yu Kawakami , Daisuke Nakajo
We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain … We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover, we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.
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Recent progress in value distribution of the hyperbolic Gauss map

2010-03-19
Yu Kawakami , 裕 川上
We give a brief survey of our work on value distribution of the hyperbolic Gauss map. In particular, we define algebraic class for constant mean curvature one surfaces in the … We give a brief survey of our work on value distribution of the hyperbolic Gauss map. In particular, we define algebraic class for constant mean curvature one surfaces in the hyperbolic three-space and give a ramification estimate for the hyperbolic Gauss map of them. Moreover, we also give an effective estimate for the number of exceptional values of the hyperbolic Gauss maps of flat fronts in the hyperbolic three-space.

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

2013-08-20
Yu Kawakami
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Value distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space

2009-08-10
Yu Kawakami
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 … 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.

A Note on a Unicity Theorem for the Gauss Maps of Complete Minimal Surfaces in Euclidean Four-space

2017-03-10
Pham Hoang Ha , Yu Kawakami
Abstract The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. … Abstract The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.
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Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

2017-02-24
Reiko Aiyama , Kazuo Akutagawa , Satoru Imagawa +1 more
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A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

2011-10-14
Yu Kawakami
We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this … We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space.

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

2015-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

Ramification estimates for the hyperbolic Gauss map

2009-12-01
Yu Kawakami
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 … 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.

Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space

2012-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more

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

2015-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 $(2\pi/n)$-rotation symmetry with respect to its … 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 $(2\pi/n)$-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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Value distribution of the Gauss map of improper affine spheres

2010-04-09
Yu Kawakami , Daisuke Nakajo
We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain … We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.

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

2012-05-22
Yu Kawakami
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 … 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.

The Gauss map of pseudo-algebraic minimal surfaces in $R^{4}$

2006-03-14
Yu Kawakami
In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number for the Gauss map of pseudo-algebraic minimal surfaces in Euclidean four-space … In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number for the Gauss map of pseudo-algebraic minimal surfaces in Euclidean four-space and give a kind of unicity theorem.

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

2012-01-01
Yu Kawakami
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 … 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.
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Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space

2015-07-23
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

Ramification estimates for the hyperbolic Gauss map

2008-01-01
Yu Kawakami
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 … 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.

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

2009-01-01
Yu Kawakami
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 … 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.

Value distribution for the Gauss maps of various classes of surfaces

2020-10-13
Yu Kawakami
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal … We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine spheres in the affine 3-space, and flat surfaces in hyperbolic 3-space. In particular, we elucidate the geometric background of their results.
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Function-theoretic properties for the Gauss maps of various classes of surfaces

2013-11-08
Yu Kawakami
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 … 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.

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

2016-06-08
Reiko Aiyama , Kazuo Akutagawa , Satoru Imagawa +1 more
We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number … We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.

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

2015-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

Heinz-type mean curvature estimates in Lorentz-Minkowski space

2020-10-08
Atsufumi Honda , Yu Kawakami , Miyuki Koiso +1 more
Abstract We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a … Abstract We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.
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Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

2022-07-06
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +4 more
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Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

2016-01-01
Reiko Aiyama , Kazuo Akutagawa , Satoru Imagawa +1 more
We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number … We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.
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Value distribution for the Gauss maps of various classes of surfaces

2017-01-01
Yu Kawakami
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal … We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine spheres in the affine 3-space and flat surfaces in hyperbolic 3-space. In particular, we elucidate the geometric background of their results.
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CMC-1 trinoids in H^3 and metrics of constant curvature one with conical singularities on S^2

2010-08-23
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

The Gauss map of pseudo-algebraic minimal surfaces in $\mathbf{R}^{4}$

2006-01-01
Yu Kawakami
In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number for the Gauss map of pseudo-algebraic minimal surfaces in Euclidean four-space … In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number for the Gauss map of pseudo-algebraic minimal surfaces in Euclidean four-space and give a kind of unicity theorem.

The Gauss map and total curvature of complete minimal Lagrangian surfaces in the complex two-space

2014-01-01
Reiko Aiyama , Kazuo Akutagawa , Yu Kawakami
The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the … The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagrangian surface with finite total curvature in the complex two-space. Moreover, we prove that if the Gauss map of a complete minimal Lagrangian surface which is not a Lagrangian plane omits three values, then it takes all other values infinitely many times.

Value distribution theoretical properties of the Gauss map of pseudo-algebraic minimal surfaces

2006-01-01
Yu Kawakami
In this thesis, we study value distribution theoretical properties of the Gauss map of pseudo-algebraic minimal surfaces in n-dimensional Euclidean space. After reviewing basic facts, we give estimates for the … In this thesis, we study value distribution theoretical properties of the Gauss map of pseudo-algebraic minimal surfaces in n-dimensional Euclidean space. After reviewing basic facts, we give estimates for the number of exceptional values and the totally ramified value numbers and the corresponding unicity theorems for them.

Value distribution for the Gauss maps of various classes of surfaces

2017-07-12
Yu Kawakami
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal … We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine spheres in the affine 3-space and flat surfaces in hyperbolic 3-space. In particular, we elucidate the geometric background of their results.

A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space

2016-12-14
Pham Hoang Ha , Yu Kawakami
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. … The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

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

2010-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.
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Value distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space

2012-01-01
Yu Kawakami , 裕 川上
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 … 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. Introduction The study of flat surfaces in the hyperbolic 3-space H3 has made a great advance in the last decade. Indeed, Galvez, Martinez and Milan [GMM] established a Weierstrasstype representation formula for such surfaces. Moreover, Kokubu, Umehara and Yamada ([KUY1], [KUY2]) investigated global properties of flat surfaces in H3 with certain kind of singularities, called flat fronts (For precise definition, see Section 1 of this paper). In particular, they gave a representation formula constructing a flat front from a given pair of hyperbolic Gauss maps and an Osserman-type inequality for complete (in the sense of [KUY2], see also Section 1 of this paper) flat fronts. More recently, Kokubu, Rossman, Saji, Umehara and Yamada [KRSUY] gave criteria for a singular point on a flat front in H3 be a cuspidal edge or swallowtail and proved the generically flat fronts in H3 admit only cuspidal edges and swallowtails. Moreover, Roitman [Ro] and Kokubu, Rossman, Umehara and Yamada [KRUY1] obtained interesting results on flat surfaces or (p-)fronts in H3 and their caustics. Furthermore, Kokubu, Rossman, Umehara and Yamada [KRUY2] also investigate the asymptotic behavior of ends of flat fronts in H3. However, we have not seen the study of value distribution of the hyperbolic Gauss maps for complete flat fronts in H3 before. On the other hand, we have recently obtained some results on value distribution of the Gauss map of complete minimal surfaces in Euclidean 3-space R and the hyperbolic Gauss map of complete constant mean curvature one (CMC-1, for short) surfaces in H3. For instance, we [Ka1] found algebraic minimal surfaces in R with totally ramified value number of the Gauss map equals 2.5 (By an algebraic minimal surface, we mean a 2000 Mathematics Subject Classification. Primary 53A10; Secondary 30D35, 53A35, 53C42.

Heinz-type mean curvature estimates in Lorentz-Minkowski space

2020-01-01
Atsufumi Honda , Yu Kawakami , Miyuki Koiso +1 more
We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, … We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.
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A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space

2016-01-01
Pham Hoang Ha , Yu Kawakami
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. … The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.
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Function-theoretic properties for the Gauss maps of various classes of surfaces

2013-01-01
Yu Kawakami
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 … 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.

Value distribution of the Gauss map of improper affine spheres

2010-01-01
Yu Kawakami , Daisuke Nakajo
We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain … We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.
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A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

2011-01-01
Yu Kawakami
We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this … We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space.
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Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

2020-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +4 more
We show that a certain simply-stated notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic extension. We also give a useful … We show that a certain simply-stated notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic extension. We also give a useful criterion for that notion of analytic completeness by defining arc-properness of continuous maps, which can be considered as a very weak version of properness. As an application, we judge the analytic completeness of a certain class of constant mean curvature surfaces (the so-called "G-catenoids") or their analytic extensions in the de Sitter 3-space.

The Gauss images of complete minimal surfaces of genus zero of finite total curvature

2023-01-01
Yu Kawakami , Watanabe Mototsugu
This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We … This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of a number of totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of a number of totally ramified values of their Gauss maps are 2.5 in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study.

Bloch-Ros principle and its application to surface theory

2024-02-20
Shunsuke Kasao , Yu Kawakami
There exists the duality between normal family theory and value distribution theory of meromorphic functions, which is called the Bloch principle. Zalcman formulated a more precise statement on it. In … There exists the duality between normal family theory and value distribution theory of meromorphic functions, which is called the Bloch principle. Zalcman formulated a more precise statement on it. In this paper, based on the Zalcman and Ros work, we comprehend the phenomenon of the trinity among normal family theory, value distribution theory and minimal surface theory and give a systematic description to the relationship among the Montel theorem, the Liuoville theorem and the Bernstein theorem as well as the Carath\'{e}odory-Montel theorem, the Picard little theorem and the Fujimoto theorem. We call this phenomenon Bloch-Ros principle. We also generalize the Bloch-Ros principle to various classes of surfaces, for instance, maxfaces in the Lorentz-Minkowski $3$-space, improper affine fronts in the affine $3$-space and flat fronts in the hyperbolic $3$-space. In particular, we give an effective criterion to determine which properties for meromorphic functions that play a role of the Gauss maps of these classes of surfaces satisfy the Gaussian curvature estimate.
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The Gauss Images of Complete Minimal Surfaces of Genus Zero of Finite Total Curvature

2024-06-25
Yu Kawakami , Watanabe Mototsugu
Abstract This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. … Abstract This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of the totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of the totally ramified values of their Gauss maps are $$5/2\,(=2.5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2.5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study.
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Surfaces with concentric or parallel K-contours

2024-05-05
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu

Surfaces with concentric or parallel $K$-contours

2024-04-21
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu
Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given. Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.
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A survey on Bernstein-type theorems for entire graphical surfaces

2024-03-21
Yu Kawakami
We survey Bernstein-type theorems of graphical surfaces in the Euclidean space and the Lorentz-Minkowski space. More specifically, we explain several proofs of the Bernstein theorem for minimal graphs in the … We survey Bernstein-type theorems of graphical surfaces in the Euclidean space and the Lorentz-Minkowski space. More specifically, we explain several proofs of the Bernstein theorem for minimal graphs in the Euclidean 3-space. Furthermore, we show the Heinz-type mean curvature estimates for graphs in the Euclidean 3-space and space-like graphs in the Lorentz-Minkowski 3-space. As an application of these estimates, we give Bernstein-type theorems for constant mean curvature graphs in the Euclidean 3-space and constant mean curvature space-like graphs in the Lorentz-Minkowski 3-space, respectively. We also study Bernstein-type results for minimal graphs in the Euclidean 4-space and the Calabi-Bernstein theorem in the Lorentz-Minkowski 3-space.
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Bloch-Ros principle and its application to surface theory

2024-02-20
Shunsuke Kasao , Yu Kawakami
There exists the duality between normal family theory and value distribution theory of meromorphic functions, which is called the Bloch principle. Zalcman formulated a more precise statement on it. In … There exists the duality between normal family theory and value distribution theory of meromorphic functions, which is called the Bloch principle. Zalcman formulated a more precise statement on it. In this paper, based on the Zalcman and Ros work, we comprehend the phenomenon of the trinity among normal family theory, value distribution theory and minimal surface theory and give a systematic description to the relationship among the Montel theorem, the Liuoville theorem and the Bernstein theorem as well as the Carath\'{e}odory-Montel theorem, the Picard little theorem and the Fujimoto theorem. We call this phenomenon Bloch-Ros principle. We also generalize the Bloch-Ros principle to various classes of surfaces, for instance, maxfaces in the Lorentz-Minkowski $3$-space, improper affine fronts in the affine $3$-space and flat fronts in the hyperbolic $3$-space. In particular, we give an effective criterion to determine which properties for meromorphic functions that play a role of the Gauss maps of these classes of surfaces satisfy the Gaussian curvature estimate.
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The Gauss images of complete minimal surfaces of genus zero of finite total curvature

2023-01-01
Yu Kawakami , Watanabe Mototsugu
This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We … This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of a number of totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of a number of totally ramified values of their Gauss maps are 2.5 in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study.

Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

2022-07-06
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +4 more
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Value distribution for the Gauss maps of various classes of surfaces

2020-10-13
Yu Kawakami
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal … We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine spheres in the affine 3-space, and flat surfaces in hyperbolic 3-space. In particular, we elucidate the geometric background of their results.
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Heinz-type mean curvature estimates in Lorentz-Minkowski space

2020-10-08
Atsufumi Honda , Yu Kawakami , Miyuki Koiso +1 more
Abstract We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a … Abstract We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.
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Heinz-type mean curvature estimates in Lorentz-Minkowski space

2020-01-01
Atsufumi Honda , Yu Kawakami , Miyuki Koiso +1 more
We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, … We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.
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Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

2020-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +4 more
We show that a certain simply-stated notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic extension. We also give a useful … We show that a certain simply-stated notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic extension. We also give a useful criterion for that notion of analytic completeness by defining arc-properness of continuous maps, which can be considered as a very weak version of properness. As an application, we judge the analytic completeness of a certain class of constant mean curvature surfaces (the so-called "G-catenoids") or their analytic extensions in the de Sitter 3-space.

Value distribution for the Gauss maps of various classes of surfaces

2017-07-12
Yu Kawakami
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal … We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine spheres in the affine 3-space and flat surfaces in hyperbolic 3-space. In particular, we elucidate the geometric background of their results.

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

2017-04-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 $(2\pi/n)$-rotation symmetry with respect to its … 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 $(2\pi/n)$-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.

A Note on a Unicity Theorem for the Gauss Maps of Complete Minimal Surfaces in Euclidean Four-space

2017-03-10
Pham Hoang Ha , Yu Kawakami
Abstract The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. … Abstract The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.
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Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

2017-02-24
Reiko Aiyama , Kazuo Akutagawa , Satoru Imagawa +1 more
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Value distribution for the Gauss maps of various classes of surfaces

2017-01-01
Yu Kawakami
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal … We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine spheres in the affine 3-space and flat surfaces in hyperbolic 3-space. In particular, we elucidate the geometric background of their results.
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A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space

2016-12-14
Pham Hoang Ha , Yu Kawakami
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. … The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Entire zero-mean curvature graphs of mixed type in Lorentz–Minkowski 3-space

2016-10-18
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

2016-06-08
Reiko Aiyama , Kazuo Akutagawa , Satoru Imagawa +1 more
We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number … We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

2016-01-01
Reiko Aiyama , Kazuo Akutagawa , Satoru Imagawa +1 more
We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number … We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.
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A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space

2016-01-01
Pham Hoang Ha , Yu Kawakami
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. … The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.
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Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space

2015-07-23
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

2015-02-14
Yu Kawakami
Abstract 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, … Abstract 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.
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Zero mean curvature entire graphs of mixed type in Lorentz-Minkowski 3-space

2015-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

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

2015-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 $(2\pi/n)$-rotation symmetry with respect to its … 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 $(2\pi/n)$-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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Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space

2015-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

The Gauss map and total curvature of complete minimal Lagrangian surfaces in the complex two-space

2014-01-01
Reiko Aiyama , Kazuo Akutagawa , Yu Kawakami
The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the … The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagrangian surface with finite total curvature in the complex two-space. Moreover, we prove that if the Gauss map of a complete minimal Lagrangian surface which is not a Lagrangian plane omits three values, then it takes all other values infinitely many times.

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

2013-11-08
Yu Kawakami
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 … 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.

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

2013-08-20
Yu Kawakami
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Function-theoretic properties for the Gauss maps of various classes of surfaces

2013-01-01
Yu Kawakami
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 … 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.

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

2012-12-06
Yu Kawakami
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Value distribution of the Gauss map of improper affine spheres

2012-07-01
Yu Kawakami , Daisuke Nakajo
We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain … We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover, we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.
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On the maximal number of exceptional values of Gauss maps for various classes of surfaces

2012-05-22
Yu Kawakami
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 … 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.

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

2012-01-01
Yu Kawakami , 裕 川上
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 … 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. Introduction The study of flat surfaces in the hyperbolic 3-space H3 has made a great advance in the last decade. Indeed, Galvez, Martinez and Milan [GMM] established a Weierstrasstype representation formula for such surfaces. Moreover, Kokubu, Umehara and Yamada ([KUY1], [KUY2]) investigated global properties of flat surfaces in H3 with certain kind of singularities, called flat fronts (For precise definition, see Section 1 of this paper). In particular, they gave a representation formula constructing a flat front from a given pair of hyperbolic Gauss maps and an Osserman-type inequality for complete (in the sense of [KUY2], see also Section 1 of this paper) flat fronts. More recently, Kokubu, Rossman, Saji, Umehara and Yamada [KRSUY] gave criteria for a singular point on a flat front in H3 be a cuspidal edge or swallowtail and proved the generically flat fronts in H3 admit only cuspidal edges and swallowtails. Moreover, Roitman [Ro] and Kokubu, Rossman, Umehara and Yamada [KRUY1] obtained interesting results on flat surfaces or (p-)fronts in H3 and their caustics. Furthermore, Kokubu, Rossman, Umehara and Yamada [KRUY2] also investigate the asymptotic behavior of ends of flat fronts in H3. However, we have not seen the study of value distribution of the hyperbolic Gauss maps for complete flat fronts in H3 before. On the other hand, we have recently obtained some results on value distribution of the Gauss map of complete minimal surfaces in Euclidean 3-space R and the hyperbolic Gauss map of complete constant mean curvature one (CMC-1, for short) surfaces in H3. For instance, we [Ka1] found algebraic minimal surfaces in R with totally ramified value number of the Gauss map equals 2.5 (By an algebraic minimal surface, we mean a 2000 Mathematics Subject Classification. Primary 53A10; Secondary 30D35, 53A35, 53C42.

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

2012-01-01
Yu Kawakami
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 … 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.
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Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space

2012-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

2011-10-14
Yu Kawakami
We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this … We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space.

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

2011-08-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
CMC-1 trinoids (i.e. constant mean curvature one immersed surfaces of genus zero with three regular embedded ends) in hyperbolic 3-space $H^{3}$ are irreducible generically, and the irreducible ones have been … CMC-1 trinoids (i.e. constant mean curvature one immersed surfaces of genus zero 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 here we give an explicit description of CMC-1 trinoids in $H^{3}$ that includes the reducible case.
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A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

2011-01-01
Yu Kawakami
We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this … We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space.
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CMC-1 trinoids in H^3 and metrics of constant curvature one with conical singularities on S^2

2010-08-23
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.

Value distribution of the Gauss map of improper affine spheres

2010-04-09
Yu Kawakami , Daisuke Nakajo
We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain … We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.

Recent progress in value distribution of the hyperbolic Gauss map

2010-03-19
Yu Kawakami , 裕 川上
We give a brief survey of our work on value distribution of the hyperbolic Gauss map. In particular, we define algebraic class for constant mean curvature one surfaces in the … We give a brief survey of our work on value distribution of the hyperbolic Gauss map. In particular, we define algebraic class for constant mean curvature one surfaces in the hyperbolic three-space and give a ramification estimate for the hyperbolic Gauss map of them. Moreover, we also give an effective estimate for the number of exceptional values of the hyperbolic Gauss maps of flat fronts in the hyperbolic three-space.

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

2010-01-01
Shoichi Fujimori , Yu Kawakami , Masatoshi Kokubu +3 more
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 … 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.
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Value distribution of the Gauss map of improper affine spheres

2010-01-01
Yu Kawakami , Daisuke Nakajo
We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain … We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.
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Ramification estimates for the hyperbolic Gauss map

2009-12-01
Yu Kawakami
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 … 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.

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

2009-08-10
Yu Kawakami
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 … 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.

The Gauss map of pseudo‐algebraic minimal surfaces in R<sup>4</sup>

2009-01-20
Yu Kawakami
Abstract In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number of the Gauss map for pseudo‐algebraic and algebraic minimal surfaces … Abstract In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number of the Gauss map for pseudo‐algebraic and algebraic minimal surfaces in Euclidean four‐space and give a kind of unicity theorem (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

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

2009-01-01
Yu Kawakami
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 … 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.

Ramification estimates for the hyperbolic Gauss map

2008-01-01
Yu Kawakami
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 … 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.

The Gauss map of pseudo-algebraic minimal surfaces

2008-01-01
Yu Kawakami , Ryoichi Kobayashi , Reiko Miyaoka
We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally … We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a new proof of Fujimoto's theorem for this class, which not only simplifies the proof but also reveals the geometric meaning behind it.
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Coauthor Papers Together
Shoichi Fujimori 14
Masatoshi Kokubu 14
Masaaki Umehara 12
Wayne Rossman 12
Kotaro Yamada 11
Kazuo Akutagawa 4
Reiko Aiyama 4
Daisuke Nakajo 3
Satoru Imagawa 3
Pham Hoang Ha 3
Watanabe Mototsugu 2
Miyuki Koiso 2
Atsufumi Honda 2
裕 川上 2
Syunsuke Tori 2
Koshi Yamada 1
Shunsuke Kasao 1
Shang‐Da Yang 1
Ryoichi Kobayashi 1
Seong-Deog Yang 1
Reiko Miyaoka 1