Therefore F 0 is a plane and F 1 is a horosphere. By Theorem 4 of Jorge and Meeks , each end of F 0 is embedded if and only if. Since any complete minimal surface with finite total curvature and one embedded end is a plane, and since the only complete minimal surface in with finite total curvature and two embedded ends is the catenoid Schoen , F 0 M is a torus with three embedded planar ends.

But Theorem 26 of Kusner and Schmitt showed that such a surface does not exist, completing the proof. Then F 0 and F 1 each have at least 4 ends. Theorem 3 of Miyaoka and Sato found examples of complete minimal surfaces of genus one with four embedded ends, but they all contain non-planar ends.

Costa and Kusner and Schmitt found examples of complete minimal surfaces of genus one with four embedded planar ends. But none of them satisfies the condition that the Hopf differential extends holomorphically to the ends. Defining annular ends to be those which are homeomorphic to punctured disks, Theorem 12 of Collin et al.

In particular, such a surface does not have horosphere ends. We saw in the proof of Proposition 3, in conjunction with Remark 4, that any single embedded annular end asymptotic to a CMC 1 catenoid in cannot have a corresponding minimal cousin in with an embedded end.

## Translation surfaces generated by Mannheim curves in three dimensional Euclidean space

So we have the following corollary, in which we do not need to assume that M has finite topology, since finite topology was not assumed in Theorem 12 of Collin et al. Let M be a Riemann surface. Assume that F 1 and F 0 are cousins. In fact, each end of F 1 in this example is an n -fold cover of an embedded end, and F 0 is an embedding.

In fact, in this example, each end of F 1 is embedded, and each end of F 0 is an n -fold cover of an embedded end. Theorem 3.

## Coulomb Frames in the Normal Bundle of Surfaces in Euclidean Spaces

Corollary 3. Furthermore, we allow M to have positive genus, so we are not considering well-definedness merely on domains which are simply-connected or homeomorphic to. A rigidity theorem for properly embedded minimal surfaces in R3, J Diff Geom The geometry of finite topology Bryant surfaces, Ann of Math Complete minimal surfaces in R3 of genus one and four planar embedded ends, Proc of Amer Math Soc The topology of complete minimal surfaces of finite total Gaussian curvature, Topology, The spinor representation of minimal surfaces, G.

Complete minimal surfaces in S 3 , Ann of Math On complete minimal surfaces whose Gauss maps miss two directions, Arch Math Lectures on minimal surfaces, Vol.

## On the minimal surfaces in Euclidean space with density | Nonlinear Studies

Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus, Tohoku Math J On the geometry of constant mean curvature one surfaces in hyperbolic space, Illinois J Math Uniqueness, symmetry and embeddedness of minimal surfaces, J Diff Geom A parametrization of the Weierstrass formulae and perturbation of complete minimal surfaces in R3 into the hyperbolic 3-space, J Reine Angew Math Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space, Ann of Math Surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends, Tohoku Math J Correspondence to Shoichi Fujimori E-mail: fujimori math.

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