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Quadratic tilt-excess decay and strong maximum principle for varifolds

Reiner Schätzle (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we prove that integral n -varifolds μ in codimension 1 with H μ ∈ L loc p ( μ ) , p > n , p ≥ 2 have quadratic tilt-excess decay tiltex μ ( x , ϱ , T x μ ) = O x ( ϱ 2 ) for μ -almost all x , and a strong maximum principle which states that these varifolds cannot be touched by smooth manifolds whose mean curvature is given by the weak mean curvature H μ , unless the smooth manifold is locally contained in the support of μ .

Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space

Georgi Ganchev, Velichka Milousheva (2014)

Open Mathematics

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.

Ramification of the Gauss map of complete minimal surfaces in ℝ 3 and ℝ 4 on annular ends

Gerd Dethloff, Pham Hoang Ha (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this article, we study the ramification of the Gauss map of complete minimal surfaces in ℝ 3 and ℝ 4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement...

Real hypersurfaces with a special transversal vector field

Zuzanna Szancer (2012)

Annales Polonici Mathematici

Real affine hypersurfaces of the complex space ℂ n + 1 are studied. Some properties of the structure determined by a J-tangent transversal vector field are proved. Moreover, some generalizations of the results obtained by V. Cruceanu are given.

Real hypersurfaces with an induced almost contact structure

Michał Szancer, Zuzanna Szancer (2009)

Colloquium Mathematicae

We study real affine hypersurfaces f : M → ℂ n + 1 with an almost contact structure (φ,ξ,η) induced by any J-tangent transversal vector field. The main purpose of this paper is to show that if (φ,ξ,η) is metric relative to the second fundamental form then it is Sasakian and moreover f(M) is a piece of a hyperquadric in ℝ 2 n + 2 .

Real hypersurfaces with parallel induced almost contact structures

Zuzanna Szancer (2012)

Annales Polonici Mathematici

Real affine hypersurfaces of the complex space ℂ n + 1 with a J-tangent transversal vector field and an induced almost contact structure (φ,ξ,η) are studied. Some properties of hypersurfaces with φ or η parallel relative to an induced connection are proved. Also a local characterization of these hypersurfaces is given.

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