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12
The Spinor Representation of Minimal Surfaces
 in space, GANG Preprint III.27
, 1996
"... this paper we investigate the interplay between spin structures on a Riemann surface M and immersions of M into threespace. Here, a spin structure is a complex line bundle S over M such that ..."
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Cited by 15 (1 self)
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this paper we investigate the interplay between spin structures on a Riemann surface M and immersions of M into threespace. Here, a spin structure is a complex line bundle S over M such that
The spinor representation of surfaces in space
, 1996
"... Abstract. The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan [32], which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the canonical line bundle K = T(M). Given a conform ..."
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Cited by 11 (0 self)
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Abstract. The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan [32], which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the canonical line bundle K = T(M). Given a conformal immersion of M into R 3, the unique spin strucure on S 2 pulls back via the Gauss map to a spin structure S on M, and gives rise to a pair of smooth sections (s1, s2) of S. Conversely, any pair of sections of S generates a (possibly periodic) conformal immersion of M under a suitable integrability condition, which for a minimal surface is simply that the spinor sections are meromorphic. A spin structure S also determines (and is determined by) the regular homotopy class of the immersion by way of a Z2quadratic form qS. We present an analytic expression for the Arf invariant of qS, which decides whether or not the correponding immersion can be deformed to an embedding. The Arf invariant also turns out to be an obstruction, for example, to the existence of
GENUSONE HELICOIDS FROM A VARIATIONAL POINT OF VIEW
, 2007
"... In this paper, we prove by variational means the existence of a complete, properly embedded, genusone minimal surface in R 3 that is asymptotic to a helicoid at infinity. We also prove existence of surfaces that are asymptotic to a helicoid away from the helicoid’s axis, but that have infinitely ma ..."
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Cited by 8 (2 self)
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In this paper, we prove by variational means the existence of a complete, properly embedded, genusone minimal surface in R 3 that is asymptotic to a helicoid at infinity. We also prove existence of surfaces that are asymptotic to a helicoid away from the helicoid’s axis, but that have infinitely many handles arranged periodically
A FAMILY OF TRIPLY PERIODIC COSTA SURFACES
 PACIFIC JOURNAL OF MATHEMATICS
, 2003
"... We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves. ..."
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Cited by 5 (1 self)
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We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.
A characterisation of the HoffmanWohlgemuth surfaces in terms of their symmetries
 J. Differential Geom
"... For an embedded singly periodic minimal surface ˜ M with genus ≥ 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the HoffmanWohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of ..."
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Cited by 1 (1 self)
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For an embedded singly periodic minimal surface ˜ M with genus ≥ 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the HoffmanWohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of these surfaces. 1.
Huge Singly Periodic Minimal Surfaces in R³ And Symmetries
, 1998
"... F13.54> OE 3 ) ' where e OE 1 = 1 2 (1 \Gamma eg 2 )e!; e OE 2 = i 2 (1 + eg 2 )e!; e OE 3 = e ge!; (0.1) are holomorphic oneforms on f M satisfying 3 X i=1 j e OE j j 2 6= 0: 1 Singly Periodic Minimal Surfaces 139 140 Francisco Martin and Domingo Rodriguez We label the pair (eg; e ..."
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F13.54> OE 3 ) ' where e OE 1 = 1 2 (1 \Gamma eg 2 )e!; e OE 2 = i 2 (1 + eg 2 )e!; e OE 3 = e ge!; (0.1) are holomorphic oneforms on f M satisfying 3 X i=1 j e OE j j 2 6= 0: 1 Singly Periodic Minimal Surfaces 139 140 Francisco Martin and Domingo Rodriguez We label the pair (eg; e !) as the Weierstrass representation of f M . We can work with the quotient M = f M=T , and so we have a minimal immersion x
1.1. The surfaces 1.2. The proof
, 2005
"... Teichmüller theory and handle addition for minimal surfaces ..."
unknown title
, 903
"... Lifetime of minimal tubes and coefficients of univalent functions in a circular ring ..."
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Lifetime of minimal tubes and coefficients of univalent functions in a circular ring
On the Gauss map of embedded minimal tubes 1
, 903
"... Abstract. A surface is called a tube if its levelsets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the socalled flow vector. We study how the geometry of the Gaussian image of a higherdimensional minimal tube M i ..."
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Abstract. A surface is called a tube if its levelsets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the socalled flow vector. We study how the geometry of the Gaussian image of a higherdimensional minimal tube M is controlled by the angle α(M) between the axis and the flow vector of M. We prove that the diameter of the Gauss image of M is at least 2α(M) if the angle α(M) is positive. As a consequence we derive an estimate on the length of a twodimensional minimal tube M in terms of α(M) and the total Gaussian curvature of M.