Duality for a class of minimal surfaces in ${\bf R}^{n+1}$

Small, Anthony

doi:10.2748/tmj/1178224720

Cited by 7 publications

(8 citation statements)

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“…Suppose that (g, f ) is a pair of holomorphic functions on a Riemann surface M. The following are versions of the classical Weierstrass formulae in free form [1,3,5,9,10], and give the coordinate functions of a null holomorphic curve ψ : M * −→ C 3 ; which is to say that ψ satisfies (ψ 1 ) 2 + (ψ 2 ) 2 + (ψ 3 ) 2 = 0. Here M * is obtained from M by deleting a finite number of points, and d f /dg = f /g , d 2 f /dg 2 = (d f /dg) /g , etc.…”

Section: Introductionmentioning

confidence: 99%

Formulae for null curves deriving from elliptic curves

Small

2008

Journal of Geometry and Physics

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Section: Introductionmentioning

confidence: 99%

Formulae for null curves deriving from elliptic curves

Small

2008

Journal of Geometry and Physics

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“…The correspondence was first found by Lie and rediscovered by Hitchin, see [7] for further details and references. In 2.6 of [1], Duistermaat asks after the relationship between the Lie correspondence and the construction of his paper; we describe it here.…”

Section: Relationship With the Lie Correspondencementioning

confidence: 99%

“…Analogous statements apply to O (n). O (2)), whose Gauss map γ ψ is nonconstant, has a Gauss transform ψ : X −→ O (2); this lifts, away from the branch points of γ ψ , to˜ ψ : X * −→ Spé O (2), and ψ| X * = •˜ ψ , [7]. It is easy to show that ψ = ˜ * ψ .…”

Section: Relationship With the Lie Correspondencementioning

confidence: 99%

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Remarks on a construction of Duistermaat

Small¹

2004

MATH. SCAND.

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“…Lie discovered a duality between null curves in C 3 and 'free' curves on a singular quadric cone in P 3 , see [3], [11], [17] and [18] for further details. In fact this may be understood, following Lie, in terms of classical osculation duality between curves in P 3 and P * 3 .…”

Section: Introductionmentioning

confidence: 99%

Algebraic minimal surfaces in $\mathsf R^4$

Small¹

2004

MATH. SCAND.

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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.

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Duality for a class of minimal surfaces in ${\bf R}^{n+1}$

Cited by 7 publications

References 18 publications

Formulae for null curves deriving from elliptic curves

Formulae for null curves deriving from elliptic curves

Remarks on a construction of Duistermaat

Algebraic minimal surfaces in $\mathsf R^4$

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