We show that for an embedded minimal disk in R 3 , near points of large curvature the surface is bi-Lipschitz with a piece of a helicoid. Additionally, a simplified proof of the uniqueness of the helicoid is provided.

Abstract. In this paper we show that a complete, embedded minimal surface in R 3 , with finite topology and one end, is conformal to a once-punctured compact Riemann surface. Moreover, using this conformal structure and the embeddedness of the surface, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. More precisely, if g is the stereographic projection of the Gauss map, then in a neighborhood of the puncture, g.p/ D exp.i˛z.p/ C F .p//, where˛2 R, z D x 3 C ix 3 is a holomorphic coordinate defined in this neighborhood and F .p/ is holomorphic in the neighborhood and extends over the puncture with a zero there. As a consequence, the end is asymptotic to a helicoid. This completes the understanding of the conformal and geometric structure of the ends of complete, embedded minimal surfaces in R 3 with finite topology.Mathematics Subject Classification (2010). 53A10; 49Q05.

Abstract. We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the (n − 2)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points x ∈ X (k) with k ≤ n − 2, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of x ∈ X.

Abstract. In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in [10] by adopting the more precise and powerful version of the methodology which was developed in [14]. As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in [10] and we produce a very large class of new embedded examples of finite topology.

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