Flat fronts in hyperbolic 3space and their caustics

 Kokubu Masatoshi KOKUBU Masatoshi
 Department of Natural Science School of Engineering Tokyo Denki University

 Rossman Wayne ROSSMAN Wayne
 Department of Mathematics Faculty of Science Kobe University

 Umehara Masaaki UMEHARA Masaaki
 Department of Mathematics Graduate School of Science Osaka University

 YAMADA Kotaro
 Faculty of Mathematics Kyushu University
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Author(s)

 Kokubu Masatoshi KOKUBU Masatoshi
 Department of Natural Science School of Engineering Tokyo Denki University

 Rossman Wayne ROSSMAN Wayne
 Department of Mathematics Faculty of Science Kobe University

 Umehara Masaaki UMEHARA Masaaki
 Department of Mathematics Graduate School of Science Osaka University

 YAMADA Kotaro
 Faculty of Mathematics Kyushu University
Abstract
After Gálvez, Martínez and Milán discovered a (Weierstrasstype) holomorphic representation formula for flat surfaces in hyperbolic 3space <i>H</i><sup>3</sup>, the first, third and fourth authors here gave a framework for complete flat fronts with singularities in <i>H</i><sup>3</sup>. In the present work we broaden the notion of completeness to <i>weak completeness</i>, and of front to <i>pfront</i>. As a front is a pfront and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces. <br>This more general class contains the caustics of flat fronts  shown also to be flat by Roitman (who gave a holomorphic representation formula for them)  which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only pfronts. Using the new framework, we obtain characterizations for caustics.
Journal

 Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 59(1), 265299, 20070101
The Mathematical Society of Japan
References: 17

1
 Surfaces of mean curvature one in hyperbolic space

BRYANT R.
Theorie des varietes minimales et applications, 1988
Cited by (1)

2
 <no title>

GALVEZ J. A.
Isometric immersions of R^2 into R^4 and perturbation of Hopf tori
Cited by (1)

3
 Embedded isolated singularities of flat surfaces in hyperbolic 3space

GALVEZ J. A.
Calc. Var. Parial Differential Equations 24, 239260, 2004
Cited by (1)

4
 Flat surfaces in hyperbolic 3space

GALVEZ J. A.
Math. Ann. 316, 419435, 2000
Cited by (1)

5
 Singularities of flat fronts in hyperbolic space

KOKUBU M.
Pacific J. Math. 221, 303351, 2005
Cited by (1)

6
 An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)

KOKUBU M.
Osaka J. Math. 40(3), 697715, 2003
Cited by (1)

7
 Flat fronts in hyperbolic 3space

KOKUBU M.
Pacific J. Math. 216, 149175, 2004
Cited by (1)

8
 <no title>

OSSERMAN R.
A survey of minimal surfaces, 1986
Cited by (1)

9
 <no title>

ROITMAN P.
Flat surfaces in hyperbolic 3space as normal surfaces to a congruence of geodesics
Cited by (1)

10
 <no title>

SINGER I. M.
Lecture notes on elementary topology and geometry, 1967
Cited by (1)

11
 The geometry of fronts

SAJI K.
math. DG/0503236
Cited by (1)

12
 Les surfaces euclidiennes a singularites coniques

TROYANOV M.
Enseign. Math. (2) 32, 7994, 1986
Cited by (1)

13
 The Minding formula and its applications

FANG Y.
Arch. Math. 72(6), 473480, 1999
DOI Cited by (1)

14
 Periodicity of the asymptotic curves on flat tori in S3

KITAGAWA Y.
Journal of the Mathematical Society of Japan 40(3), p457476, 198807
Cited by (3)

15
 Application of soliton theory to the construction of pseudospherical surfaces in R^3

MELKO M.
Annals of Global Analysis and Geometry 11, 65107, 1993
DOI Cited by (1)

16
 Total curvature and minimal areas of complete open surfaces

SHIOHAMA K.
Proc. Amer. Math. Soc. 94, 310316, 1985
DOI Cited by (3)

17
 Complete surfaces of constant mean curvature1 in the hyperbolic 3space

UMEHARA M.
Ann. of Math. 137, 611638, 1993
DOI Cited by (2)
Cited by: 2

1
 Value distribution of the Gauss map of improper affine spheres

KAWAKAMI Yu , NAKAJO Daisuke
Journal of the Mathematical Society of Japan 64(3), 799821, 20120701

2
 Asymptotic behavior of flat surfaces in hyperbolic 3space

KOKUBU Masatoshi , ROSSMAN Wayne , UMEHARA Masaaki , YAMADA Kotaro
Journal of the Mathematical Society of Japan 61(3), 799852, 20090701
JSTAGE References (13) Cited by (1)