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Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

Rafael López, Esma Demir (2014)

Open Mathematics

We classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis.

Hypersurfaces with constant k -th mean curvature in a Lorentzian space form

Shichang Shu (2010)

Archivum Mathematicum

In this paper, we study n ( n ≥ 3 ) -dimensional complete connected and oriented space-like hypersurfaces M n in an (n+1)-dimensional Lorentzian space form M 1 n + 1 ( c ) with non-zero constant k -th ( k < n ) mean curvature and two distinct principal curvatures λ and μ . We give some characterizations of Riemannian product H m ( c 1 ) × M n - m ( c 2 ) and show that the Riemannian product H m ( c 1 ) × M n - m ( c 2 ) is the only complete connected and oriented space-like hypersurface in M 1 n + 1 ( c ) with constant k -th mean curvature and two distinct principal curvatures, if the multiplicities of...

Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Mouhamed Moustapha Fall, Fethi Mahmoudi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Given a domain Ω of ℝ m + 1 and a k -dimensional non-degenerate minimal submanifold K of ∂ Ω with 1 ≤ k ≤ m - 1 , we prove the existence of a family of embedded constant mean curvature hypersurfaces in Ω which as their mean curvature tends to infinity concentrate along K and intersecting ∂ Ω perpendicularly along their boundaries.

Invariance groups of relative normals

Thomas Binder, Martin Wiehe (2005)

Banach Center Publications

We investigate a two-parameter family of relative normals that contains Manhart's one-parameter family and the centroaffine normal. The invariance group of each of these normals is classified, and variational problems are studied. The results are Euler-Lagrange equations for the hypersurfaces that are critical with respect to the area functionals of the induced and semi-Riemannian volume forms and a classification of the critical hyperovaloids in the two-parameter family.

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