Constant mean curvature one surfaces in hyperbolic 3-space using the Bianchi-Calò method

Lima, Levi Lopes de; Roitman, Pedro

doi:10.1590/s0001-37652002000100002

Cited by 13 publications

(7 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Instead of the usual Bryant representation formula, we use the BianchiCalò method to represent a Bryant surface which is homeomorphic to a region in C [6]. Let f = f (z) be a holomorphic map defined in a region Ω ⊂ C, and let…”

Section: Bianchi-calò Methodsmentioning

confidence: 99%

See 1 more Smart Citation

On the Symmetry of Annular Bryant Surface With Constant Contact Angle

Park

2014

Korean Journal of Mathematics

View full text Add to dashboard Cite

show abstract

Section: Bianchi-calò Methodsmentioning

confidence: 99%

“…Clearly, ∂ ∞ H 3 is one of the two envelopes of S f (z) . The second envelope gives a Bryant surface whose gauss map is f [6].…”

Section: Bianchi-calò Methodsmentioning

confidence: 99%

On the Symmetry of Annular Bryant Surface With Constant Contact Angle

Park

2014

Korean Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…Small in [11] with techniques of algebraic geometry. However, in implicit form it can be found in the classical book [2] of L. Bianchi where its derivation was based on geometric ideas concerning congruences of spheres and the rolling of pairs of isometric 232 H. GOLLEK surfaces in R 3 on each other (see also [10]). The following algebraic formula is quoted from [11]:…”

Section: Given a Null Curve F =mentioning

confidence: 99%

“…• Finally, a modification of A. Small's formula (see [11], [10] or [13]) leads to an algebraic formula BS : (f , g) ∈ A Σ × A Σ → N Σ depending on two arguments. For this purpose a partial operation between null curves is defined, assigning to any pair (F 1 , F 2 ) of null curves such that their Bryant data g 1 and g 2 (i.e., the first arguments of (1.1)) are equal, a new null curve BS f ,g = F 1 (F 2 ) −1 .…”

mentioning

confidence: 99%

Algebraic representation formulas for null curves in Sl(2,C)

Gollek

2005

PDEs, Submanifolds and Affine Differential Geometry

View full text Add to dashboard Cite

Abstract. We study curves in Sl(2, C) whose tangent vectors have vanishing length with respect to the biinvariant conformal metric induced by the Killing form, so-called null curves. We establish differential invariants of them that resemble infinitesimal arc length, curvature and torsion of ordinary curves in Euclidean 3-space. We discuss various differential-algebraic representation formulas for null curves. One of them, a modification of the Bianchi-Small formula, gives an Sl(2, C)-equivariant bijection between pairs of meromorphic functions and null curves. The inverse of this formula is also differential-algebraic. The other one is based on an integral formula deduced from that of R. Bryant, using certain natural differential operators on Riemannian surfaces that we introduced in [7] for differential-algebraic representation formulas of curves in C 3 . We demonstrate some commands of a Mathematica package that resulted from our investigations, containing algebraic and graphical utilities to handle null curves, their invariants, representation formulas and associated surfaces of constant mean curvature 1 in H 3 , taking into consideration several models of H 3 .

show abstract

“…In §6 we explain how to derive this duality from the correspondence for null curves in C 4 . For other recent derivations, see [4], [7] and [8]. (For basic information about constant mean curvature 1 surfaces in H 3 , in addition to [1], one should consult the seminal papers of Umehara and Yamada and their coauthors.…”

Section: Introductionmentioning

confidence: 99%

Algebraic minimal surfaces in $\mathsf R^4$

Small¹

2004

MATH. SCAND.

View full text Add to dashboard Cite

There exists a natural correspondence between null curves in C 4 and 'free' curves on O (1) ⊕ O (1) −→ P 1 ; it underlies the existence of 'Weierstrass type formulae' for minimal surfaces in R 4 . The construction determines correspondences for minimal surfaces in R 3 , and constant mean curvature 1 surfaces in H 3 ; moreover it facilitates the study of symmetric minimal surfaces in R 4 . IntroductionOur purpose here is to describe a natural correspondence between null holomorphic curves in C 4 , and 'free' holomorphic curves on the total space of the holomorphic vector bundle O (1) ⊕ O (1) over P 1 . Our main interest in the former derives from the fact that any minimal surface in R 4 may be described as the real part of such a curve. The correspondence is particularly useful in the study of algebraic minimal surfaces, that is, the real parts of null meromorphic curves in C 4 . The construction is closely related to the classical Klein correspondence between lines in P 3 and points of the quadric Q 4 ⊂ P 5 . In fact, compactifying C 4 to Q 4 , and O (1) ⊕ O (1) to P 3 , it may be understood in terms of classical osculation duality, cf.[10]. Here we work in the 'uncompactified picture'. We feel that this makes the differential geometry clearer, in particular the appearance of nullity, and moreover eases the discussion of the relationship with the other correspondences described below.The correspondence underlies the Weierstrass-type formulae (10)- (13) below. These were found by Montcheuil [15], and studied at length by Eisenhart in [5] and [6]. The geometrical structure underlying the formulae was first exposed by Shaw [21]. This was framed in 'twistor terminology', in terms of the Klein construction. Unfortunately, this interesting paper has largely been * The author is grateful to John Denham for helpful conversations. He is indebted to the Mathematics Faculty of the University of Southampton, England, for its generous hospitality during the academic year [2001][2002], and thanks Victor Snaith for his invitation. He also thanks the Mathematics Section of the ICTP, Trieste, Italy, for its hospitality during February and March of 2002.

show abstract

Constant mean curvature one surfaces in hyperbolic 3-space using the Bianchi-Calò method

Cited by 13 publications

References 3 publications

On the Symmetry of Annular Bryant Surface With Constant Contact Angle

On the Symmetry of Annular Bryant Surface With Constant Contact Angle

Algebraic representation formulas for null curves in Sl(2,C)

Algebraic minimal surfaces in $\mathsf R^4$

Contact Info

Product

Resources

About