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In this paper, history of reserches for minimal immersions from constant Gaussian curvature 2-manifolds into space forms is explained with special emphasis of works of O. Borůvka. Then recent results for the corresponding probrem to classify minimal immersions of such surfaces in complex space forms are discussed.

Open book structures and unicity of minimal submanifolds

R. Hardt, Harold Rosenberg (1990)

Annales de l'institut Fourier

We prove unicity of certain minimal submanifolds, for example Clifford annuli in $S^{3}$ . The idea is to consider the placement of the submanifold with respect to the (singular) foliation of $S^{3}$ by the Clifford annuli whose boundary are two fixed great circles a distance $π / 2$ apart.

Optimal inequalities for embedded space-times

Stefan Haesen (2005)

Kragujevac Journal of Mathematics

Orientability of higher order Grassmannians

Michal Krupka (1994)

Mathematica Slovaca

Parabolicity and rigidity of spacelike hypersurfaces immersed in a Lorentzian Killing warped product

Eudes L. de Lima, Henrique F. de Lima, Eraldo A. Jr. Lima, Adriano A. Medeiros (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we extend a technique due to Romero et al. establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed into a Lorentzian Killing warped product whose Riemannian base has parabolic universal Riemannian covering. As applications, we obtain rigidity results concerning these hypersurfaces. A particular study of entire Killing graphs is also made.

Parallel and totally geodesic hypersurfaces of 5-dimensional 2-step homogeneous nilmanifolds

Mehri Nasehi (2016)

Czechoslovak Mathematical Journal

In this paper we study parallel and totally geodesic hypersurfaces of two-step homogeneous nilmanifolds of dimension five. We give the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces. Moreover, we prove that two-step homogeneous nilmanifolds of dimension five which have one-dimensional centre never admit parallel hypersurfaces. Also we prove that the only two-step homogeneous nilmanifolds of dimension five which admit totally geodesic hypersurfaces...

Parallel and totally geodesic hypersurfaces of solvable Lie groups

Mehri Nasehi (2016)

Archivum Mathematicum

In this paper we consider special examples of homogeneous spaces of arbitrary odd dimension which are given in [5] and [16]. We obtain the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces in both Riemannian and Lorentzian cases.

Parallel hypersurfaces

Barbara Opozda, Udo Simon (2014)

Annales Polonici Mathematici

We investigate parallel hypersurfaces in the context of relative hypersurface geometry, in particular including the cases of Euclidean and Blaschke hypersurfaces. We describe the geometric relations between parallel hypersurfaces in terms of deformation operators, and we apply the results to the parallel deformation of special classes of hypersurfaces, e.g. quadrics and Weingarten hypersurfaces.

Periodic minimal surfaces.

Brian Smyth, Tadashi Nagano (1978)

Commentarii mathematici Helvetici

p-harmonic maps and minimal submanifolds.

Paul Baird, Sigmundur Gudmundsson (1992)

Mathematische Annalen

Planar Geodesic Immersions in Pseudo-Euclidean Space.

Carol Blomstrom (1986)

Mathematische Annalen

Planar normal sections on the natural embedding of a real flag manifold.

Sánchez, Cristián U., García, Alicia N., Dal Lago, Walter (2000)

Beiträge zur Algebra und Geometrie

Plongement isométrique de la sphère à courbure non négative dans $R^{3}$

C. Zuily (1993/1994)

Séminaire Équations aux dérivées partielles (Polytechnique)

Plongements radiaux $S^{n} ↪ R^{n + 1}$ à courbure de Gauss positive prescrite

Ph. Delanoë (1985)

Annales scientifiques de l'École Normale Supérieure

Pointed $k$ -surfaces

Graham Smith (2006)

Bulletin de la Société Mathématique de France

Let $S$ be a Riemann surface. Let $ℍ^{3}$ be the $3$ -dimensional hyperbolic space and let $\partial_{\infty} ℍ^{3}$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $ϕ : S \to \partial_{\infty} ℍ^{3} = \hat{ℂ}$ . If $i : S \to ℍ^{3}$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\hat{ı}$ , of $i$ by $\hat{ı} = N$ . Let $\vec{n} : U ℍ^{3} \to \partial_{\infty} ℍ^{3}$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S, ϕ)$ is a convex immersion $i$ of constant Gaussian curvature equal to $k \in (0, 1)$ such that the Gauss lifting $(S, \hat{ı})$ is complete and $\vec{n} \circ \hat{ı} = ϕ$ . In this paper, we show...

Pointwise curvature estimates for F-stable hypersurfaces

Sven Winklmann (2005)

Annales de l'I.H.P. Analyse non linéaire

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