arXiv research

Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.

Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.

Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.

Introduces hyperbolic generalized framed surfaces and their properties.

We define the notions of $S_t^1\times S_s^1$-valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of s…

On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must b…

Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.

The paper studies helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space.

Paper proves uniqueness of specific spacetime surfaces in a lightcone.

Past lightcones of certain points in globally hyperbolic spacetimes determine the entire spacetime.

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

Proves estimate similar to De Lellis-Müller on Minkowski lightcone.

Study on stability of surfaces in null cones under area-preserving variations.

Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For …

New perspective on Ricci flow on spheres using Minkowski spacetime.

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…

Paper proves isoperimetric inequality for Minkowski spacetime.

To study spacelike surfaces in the Lorentz-Minkowski space $\Bbb R^{4}_1,$ we construct a pair of maps whose values are in the lightcone, called $\mathfrak l_r^{\pm}$-Gauss maps. We can use these maps to study umbilical spacelike surfaces and find parametrizations of spacelike surfaces of revolution of hyperbolic and …

We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observation…

We introduce the totally absolute lightcone curvature for a spacelike submanifold with general codimension and investigate global properties of this curvature. One of the consequences is that the Chern-Lashof type inequality holds. Then the notion of lightlike tightness is naturally induced.

We prove effective uniformization for nearly round 2-spheres and investigate their stability.

Bertrand framed surfaces defined in Euclidean 3-space with applications.

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isotherm…

The study characterizes surfaces with specific harmonic properties in pseudo-conformal geometry.

Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.

Study on singular points of translation surfaces under linearly dependent conditions.

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…

Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.

In this paper we introduce the framed pure braid group on $n$ strands of an oriented surface, a topological generalisation of the pure braid group $P_n$. We give different equivalents definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of $P_n$, usually…

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

Discussing moving frames for curve and surface invariants.

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

Study geometry of surfaces glued along a curve.

Introduces $\mathbb{K}$-framings for surfaces, generalizing quadratic forms.

Study on framed surfaces with bounds on Morse index.

New flow generates surfaces with constant curvature.

The problem of quasilocal energy has been extensively studied mainly in four dimensions. Here we report results regarding the quasilocal energy in spacetime dimension $n\geq 4$. After generalising three distinct quasilocal energy definitions to higher dimensions under appropriate assumptions, we evaluate their small sp…

In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the r…

The paper studies geometric properties of soliton surfaces using an extended Darboux frame field.

Any 2-dim Riemannian manifold with spherical topology can be embedded isometrically into a lightcone of the Minkowski spacetime. We apply this fact to give a proof of the Kazdan-Warner identity.

We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.

In this paper, we investigate the ruled surfaces generated by a straight line according to rotation minimizing frame (RMF). Using this frame of a straight line, we obtained the necessary and sufficient conditions when the ruled surface is developable. Also, we give some new results and theorems related to be the asympt…

In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim …

We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this fr…

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

Study helicoidal surfaces with singular points using frontals.

We prove a rigidity of the lightcone in Minkowski space. It is essentially the unique space endowed with a degenerate Riemannian metric, of lightlike type, and supporting an isometric non-proper action of a semi-simple group.

For a regular curve on a spacelike surface in Lorentz-Minkowski $3$-space, we have a moving frame along the curve which is called a Lorentzian Darboux frame. We introduce five special vector fields along the curve associated to the Lorentzian Darboux frame and investigate their singularities.