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45
The Geometry of Periodic Minimal Surfaces
, 1993
"... this paper we shall demonstrate a surprising relationship between the topology of a properly embedded periodic minimal surface in R ..."
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Cited by 34 (1 self)
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this paper we shall demonstrate a surprising relationship between the topology of a properly embedded periodic minimal surface in R
Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus
 Tôhoku Math. J
, 1997
"... Abstract. In this work we give a method for constructing a oneparameter family of complete CMC1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3space. We show that this oneparameter fami ..."
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Cited by 33 (12 self)
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Abstract. In this work we give a method for constructing a oneparameter family of complete CMC1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3space. We show that this oneparameter 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 oneparameter family. 1.
Construction of complete embedded selfsimilar surfaces under mean curvature flow
 Part II, Trans. Amer. Math. Soc
"... Abstract. We carry out the first main step towards the construction of new examples of complete embedded selfsimilar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of selfsimilar surfaces and desingularizing the intersection circle using an app ..."
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Cited by 20 (1 self)
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Abstract. We carry out the first main step towards the construction of new examples of complete embedded selfsimilar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of selfsimilar surfaces and desingularizing the intersection circle using an appropriately modified singly periodic Scherk surface, called the core. Using an inverse function theorem, we show that for small boundary conditions on the core, there is an embedded surface close to the core that is a solution of the equation for selfsimilar surfaces. This provides us with an adequate central piece to substitute for the intersection. 1.
The classification of doubly periodic minimal tori with parallel ends
"... 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 3dimensional real analytic manifold that reduces to the finite coverings of the examples d ..."
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Cited by 18 (7 self)
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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 3dimensional 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 1parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples. 1.
COMPLETE MINIMAL SURFACES IN R³
, 1999
"... In this paper we review some topics on the theory of complete minimal surfaces in three dimensional Euclidean space. ..."
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Cited by 15 (1 self)
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In this paper we review some topics on the theory of complete minimal surfaces in three dimensional Euclidean space.
Minimal surfaces of least total curvature and moduli spaces of plane polygonal arcs
 Geom. Funct. Anal
, 1998
"... 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 ..."
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Cited by 9 (5 self)
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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