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We show that there exists a complete minimal surface immersed into $ℝ^{3}$ which is conformally equivalent to a compact hyperelliptic Riemann surface of genus three minus one point. The end of the surface is of Enneper type and its total curvature is $- 16 π$ .

Complete minimal surfaces in R3.

Francisco J. López, Francisco Martín (1999)

Publicacions Matemàtiques

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

Complete minimal surfaces of arbitrary genus in a slab of $ℝ^{3}$

Celso J. Costa, Plinio A. Q. Simöes (1996)

Annales de l'institut Fourier

In this paper we construct complete minimal surfaces of arbitrary genus in $ℝ^{3}$ with one, two, three and four ends respectively. Furthermore the surfaces lie between two parallel planes of $ℝ^{3}$ .

Complete minimal surfaces with index one and stable constant mean curvature surfaces.

Antonio Ros, Francisco J. Lopez (1989)

Commentarii mathematici Helvetici

Complete Minimal Varieties in Hyperbolic Space.

Michael T. Anderson (1982)

Inventiones mathematicae

Complete noncompact submanifolds with flat normal bundle

Hai-Ping Fu (2016)

Annales Polonici Mathematici

Let Mⁿ (n ≥ 3) be an n-dimensional complete super stable minimal submanifold in $ℝ^{n + p}$ with flat normal bundle. We prove that if the second fundamental form A of M satisfies $\int_{M} i {| A |}^{α} < \infty$ , where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and $\int_{M} {| A |}^{d} < \infty$ , d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite $L^{α}$ -norm curvature in ℝ⁷ are considered.

Complete Non-Orientable Minimal Surfaces in ℝ 3 and Asymptotic Behavior

Antonio Alarcón, Francisco J. López (2014)

Analysis and Geometry in Metric Spaces

In this paperwe give new existence results for complete non-orientable minimal surfaces in ℝ3 with prescribed topology and asymptotic behavior

Complete nonorientable minmal surfaces in R3.

Marty Ross (1992)

Commentarii mathematici Helvetici

Complete real Kähler Euclidean hypersurfaces are cylinders

Luis A. Florit, Fangyang Zheng (2007)

Annales de l’institut Fourier

In this note we show that any complete Kähler (immersed) Euclidean hypersurface $M^{2 n} \subset ℝ^{2 n + 1}$ must be the product of a surface in $ℝ^{3}$ with an Euclidean factor $ℂ^{n - 1} ≅ ℝ^{2 n - 2}$ .

Complete real Kähler minimal submanifolds.

Marcos Dajczer, Lucio Rodríguez (1991)

Journal für die reine und angewandte Mathematik

Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures

Amalendu Ghosh (2016)

Mathematica Bohemica

We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g, \pm ω)$ with constant scalar curvature is either Einstein, or the dual field of $ω$ is Killing. Next, let $(M^{n}, g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g, \pm ω)$ . Then the Einstein-Weyl vector field $E$ (dual to the $1$ -form $ω$ ) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

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