Related papers: Space-like Surfaces in Minkowski Space $\mathbb E^…
The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a hypersurface in terms of the surface area, which is optimal on round spheres. In this paper we employ a locally constrained…
In a recent paper Jorge and Mercuri proved that the image of Gauss map of a complete non flat minimal surfaces in R3 with finite total curvature omits at most 2 points. In this work we follow their idea and prove 3a similar result for CMC-1…
In this paper, we continue the study of the Killing symmetries of a N-dimensional generalized Minkowski space, i.e. a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical…
In this note we consider the problem of local classification of Bonnet pairs of surfaces in 3-dimensional Minkowski space. We use split quaternions in a way similar to the use of quaternions in, arXiv:dg-ga/9610006, for the solution in…
We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover,…
This article explores the geometric algebra of Minkowski spacetime, and its relationship to the geometric algebra of Euclidean 4-space. Both of these geometric algebras are algebraically isomorphic to the 2x2 matrix algebra over Hamilton's…
In this paper, we investigate the geometry of compact spacelike biconservative hypersurfaces with constant scalar curvature in de Sitter space $\mathbb{S}_1^{m+1}(c)$, under some geometric constraints. Our results extend the understanding…
We consider the timelike minimal surface problem in Minkowski spacetimes and show local and global existence of such surfaces having arbitrary dimension $\geq 2$ and arbitrary co-dimension, provided they are initially close to a flat plane.
In this paper, based on the theory of surfaces in the four-dimensional Euclidean space which generalizes the theory of surfaces in three-dimensional Euclidean space, beside other results, we will give a characterization of points on…
In this article, we consider space-like surfaces in Robertson-Walker Space times $L^4_1(f,c)$ with comoving observer field $\frac{\partial}{\partial t}$. We study some problems related to such surfaces satisfying the geometric conditions…
In this paper, we study the singularities of spacelike constant mean curvature one (CMC 1) surfaces in the de Sitter 3-space. We prove the duality between generalized conelike singular points and 5/2-cuspidal edges on spacelike CMC 1…
We prove that an isometric immersion of a simply connected Riemannian surface M in four-dimensional Minkowski space, with given normal bundle E and given mean curvature vector H \in \Gamma(E), is equivalent to a normalized spinor field…
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the…
In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar…
In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$ using the framework of M\"{o}bius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed M\"obius…
We classify hypersurfaces of the Minkowski space $\L^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or…
We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with…
We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension)…
We prove that any regular domain in Minkowski space is uniquely foliated by spacelike constant mean curvature (CMC) hypersurfaces. This completes the classification of entire spacelike CMC hypersurfaces in Minkowski space initiated by Choi…
We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric…