this paper we investigate the interplay between spin structures on a Riemann surface M and immersions of M into three-space. Here, a spin structure is a complex line bundle S over M such that

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 Z2-quadratic 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

In this paper, we prove by variational means the existence of a complete, properly embedded, genus-one 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

We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.

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 one-forms 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

Teichmüller theory and handle addition for minimal surfaces

Life-time of minimal tubes and coefficients of univalent functions in a circular ring

Abstract. A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional 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 two-dimensional minimal tube M in terms of α(M) and the total Gaussian curvature of M.

Abstract not found