The Weierstra representation is a powerful tool bringing methods of complex analysis to bear on problems of existence in minimal surface theory. It is particularily powerful in constructing minimal… (More)

There exists a properly embedded minimal surface of genus one with a single end asymptotic to the end of the helicoid. This genus-one helicoid is constructed as the limit of a continuous… (More)

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane.… (More)

Starting from the hyperbolic definition of Klein’s surface we prove platonicity, derive the two classical equations W 7 = Z(Z − 1)2 between meromorphic functions and x3y+y3z+z3x = 0 between… (More)

Consider on a complex 1-dimensional torus Tλ an abelian differential of the second kind αλ. Assign to each λ the period quotient of αλ for two independent cycles on Tλ. For appropriate choices of αλ,… (More)

The main objective of this paper is to construct smooth 1-parameter families of embedded minimal surfaces in euclidean space that are invariant under a screw motion and are asymptotic to the… (More)

We give a new geometric computation for the Jacobian of the Riemann surface of genus 4 associated to Kepler’s small stellated dodecahedron. Starting with Threlfall’s description, we introduce several… (More)

In autumn 1993, in front of the MSRI in Berkeley, a marble sculpture by Helaman Ferguson called The Eightfold Way was revealed. This sculpture shows a compact Riemann surface of genus 3 with… (More)

A uniform and elementary treatment of many classical and new embedded triply periodic minimal surfaces in Euclidean space, based on a SchwarzChristoffel formula for periodic polygons in the plane, is… (More)

We construct Colding-Minicozzi limit minimal laminations in open domains in R with the singular set of C-convergence being any properly embedded C-curve. By Meeks’ C-regularity theorem, the singular… (More)