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ON THE MODULI OF CONSTANT MEAN CURVATURE CYLINDERS OF FINITE TYPE IN THE 3SPHERE
, 2008
"... We show that onesided Alexandrov embedded constant mean curvature cylinders of finite type in the 3sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3sphere are rotational. ..."
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Cited by 10 (1 self)
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We show that onesided Alexandrov embedded constant mean curvature cylinders of finite type in the 3sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3sphere are rotational.
Flows of constant mean curvature tori in the 3sphere: The equivariant case, arXiv:1011.2875
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Delaunay ends of constant mean curvature surfaces
"... Abstract. The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay su ..."
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Cited by 6 (5 self)
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Abstract. The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.
Global aspects of integrable surface geometry
 Proceedings for ”Integrable Systems and Quantum Field Theory at Peyresq, Fifth Meeting
"... The study of surfaces in 3space has certainly been pivotal in the development of differential geometry and geometric analysis. Real dimension two is of course special in a number of ways: the possible topological types are easy to describe, one can take advantage of complex analysis in dimension o ..."
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Cited by 3 (3 self)
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The study of surfaces in 3space has certainly been pivotal in the development of differential geometry and geometric analysis. Real dimension two is of course special in a number of ways: the possible topological types are easy to describe, one can take advantage of complex analysis in dimension one, and visualization of the objects of study in computer experiments allows many new conjectures to be formulated and tested. It is fair to say that nowadays surface geometry serves as a terrain where one can quickly migrate between diverse areas of mathematics, such as integrable systems, moduli spaces of connections and holomorphic bundles, surface group representations, algebraic geometry of special varieties, nonlinear variational problems, mathematical physics, and computational methods and visualization, bringing the ideas and techniques of one to bear on another. This article attempts to describe some of these topics and their relevance to classical problems in surface geometry in a conceptual manner. We maintain an informal style with the hope of leaving the reader with some impressions of the subject and a snapshot of some methods under current development. There is a long tradition of physics motivating advances in surface geometry. Early on
ON THE BJÖRLING PROBLEM FOR WILLMORE SURFACES
"... We use an isotropic harmonic map representation of Willmore surfaces to solve the analogue of Björling’s problem for such surfaces. Specifically, given a real analytic curve y0 in S3, together with the prescription of the values of the surface normal and the dual Willmore surface along the curve, ..."
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We use an isotropic harmonic map representation of Willmore surfaces to solve the analogue of Björling’s problem for such surfaces. Specifically, given a real analytic curve y0 in S3, together with the prescription of the values of the surface normal and the dual Willmore surface along the curve, lifted to the light cone in Minkowski 5space R51, we prove that there exists a unique pair of dual Willmore surfaces y and y ̂ satisfying the given values along the curve. We give explicit formulae for the generalized Weierstrass data for the surface pair. Similar results are derived for SWillmore surfaces in higher codimensions. For the three dimensional target, we use the solution to explicitly describe the Weierstrass data, in terms of geometric quantities, for all equivariant Willmore surfaces.