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Abstract. In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same symmetry properties exists for all given minimal surfaces satisfying certain conditions. The surfaces we construct in this paper are irreducible, and in the process of showing this, we also prove some results about the reducibility of surfaces. Furthermore, in the case that the surfaces are of genus 0, we are able to make some estimates on the range of the parameter for the one-parameter family. 1.

this paper we shall demonstrate a surprising relationship between the topology of a properly embedded periodic minimal surface in R

this paper, we develop Teichmuller theoretical methods to construct new minimal surfaces in E

rst major goal of this paper is to prove the existence of complete minimal surfaces of each genus p>1 which minimize the total curvature (equivalently, the degree of the Gau map) for their genus. The genus zero version of these surfaces is known

Let K be the space of properly embedded minimal tori in quotients of R3 by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that K is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in [9, 10, 15]. The degenerate limits of surfaces in K are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples. 1.

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Abstract. We describe a 3-parametric family K of properly embedded minimal tori with four parallel ends in quotients of R 3 by two independent translations, which we will call the standard examples. These surfaces generalize the examples given by Karcher, Meeks and Rosenberg in [3, 4, 7]. K can be endowed with a natural structure of a self-conjugated 3-dimensional real analytic manifold diffeomorphic to R × R

In this paper we review some topics on the theory of complete minimal surfaces in three dimensional Euclidean space.

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