Related papers: Evolution of Spacelike Curves and Special Timelike…
Several uniqueness results for non-compact complete stationary spacelike surfaces in an $n(\geq 3)$-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss…
We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and…
In this paper we study curves in Lorentz-Minkowski space $\mathbb{L}^2$ that are critical points of the moment of inertia with respect to the origin. This extends a problem posed by Euler in the Lorentzian setting. We obtain explicit…
We study rotational surfaces with constant Minkowski Gaussian curvature and rotational surfaces with constant Minkowski mean curvature in a $3$-dimensional normed space with rotationally symmetric norm. We have a generalization of the…
A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance…
In this paper, we introduce the pseudo-torsion functions along spacelike curves whose curvature vector field has isolated lightlike points in Lorentz-Minkowski 3-space, and prove the fundamental theorem. Moreover, we analyze the behavior of…
In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bf E}%_{1}^{3}$ that satisfy a linear Weingarten condition of type $\kappa_{1}=m\kappa_{2}+n$, where $m$ and $n$ are constant and $\kappa_{1}$ and $\kappa_{2}$ denote…
We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of…
We study the evolution of complete non-compact convex hypersurfaces in $\mathbb{R}^{n+1}$ by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of…
Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special…
We study evolutes and involutes of space curves. Although much of the material presented is not new and can be found in classic treatises, we believe that a modern and unified treatment, complemented with several novel observations, may be…
We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains…
We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an…
We consider a special family of 2-dimensional timelike surfaces in the Minkowski 4-space $\mathbb{R}^4_1$ which lie on rotational hypersurfaces with timelike axis and call them meridian surfaces of elliptic type. We study the following…
In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial…
In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff…
In this study, we have obtained the distribution parameter of a ruled surface generated by a straight line in Frenet trihedron moving along a timelike curve and also along another curve with the same parameter. At this time, the Frenet…
We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity…
We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first…
Spatial curvature is one of the fundamental cosmological parameters that is routinely constrained from observations. The forward modelling of observations, in particular of large-scale structure, often relies on large cosmological…