A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.
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48 results for “Lorentz Minkowski space”
Study on maximal surfaces with high genus in Lorentz-Minkowski space.
problem Existence of nonorientable maximal surfaces with high genus.
method Existence results for nonorientable maximal surfaces with high genus and one end.
result Existence of maximal surfaces with high genus in Lorentz-Minkowski space.
Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.
problem Characterizing maximal surfaces in Lorentz-Minkowski space.
method Three proofs showing correspondence to minimal surfaces in Euclidean space.
result Conjugate surface of a maximal graph over a convex domain is also a graph.
Unified estimates for mean curvature in Lorentz-Minkowski space.
problem Estimating mean curvature for space-like and time-like graphs.
method Using gradient bounds to derive Heinz-type estimates.
result Unified vanishing theorem for mean curvature of constant mean curvature graphs.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.
problem Existence and uniqueness of solutions to Hessian quotient equations in Lorentz-Minkowski space.
method Suitable settings to prove existence and uniqueness of solutions.
result Existence and uniqueness of solutions to the class of Hessian quotient equations.
The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.
problem Global properties of space-like surfaces with constant mean curvature in Lorentz-Minkowski space.
method Isothermic coordinate systems
result Global properties of space-like surfaces with constant mean curvature explored.
Constructs graphs with singularities in a special space.
problem Creating graphs with specific singularities in a unique space.
method Using Weierstrass representation for minimal surfaces.
result Constructs entire singly periodic graphs with isolated cone-like singularities.
Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.
problem Classifying ruled surfaces in Lorentz-Minkowski space.
method Examining homothetic self-similar solutions of the inverse mean curvature flow.
result Existence of two classes of non-cylindrical homothetic solitons.
The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.
problem Understanding null hypersurfaces in Lorentz-Minkowski space.
method Analyzing -complete null hypersurfaces as wave fronts in Euclidean space.
result Most null wave fronts can be realized as restrictions of certain -complete null wave fronts.
The paper studies helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space.
problem Investigating the properties and singularities of helicoidal surfaces in Lorentz-Minkowski space.
method Defining and analyzing two types of helicoidal surfaces, using diffeomorphic transformations and criteria for cusps and cuspidal edges.
result Identification theorems for the singular types of both 1-type and 2-type helicoidal surfaces.
Study surfaces with nonvanishing Gauss curvature in Lorentz-Minkowski space.
problem Characterizing surfaces with specific curvature properties in Lorentz-Minkowski space.
method Analyzing surfaces satisfying ΔIIIx = Ax, where III is the second fundamental form.
result Surfaces are either minimal or pseudospheres with real or imaginary radii.
New approach classifies rotational Weingarten surfaces in Lorentz-Minkowski space.
problem Classifying rotational Weingarten surfaces in Lorentz-Minkowski space.
method Using geometric linear momentum of generatrix curves with respect to axes of revolution.
result Unified framework for three causal types of rotation axes.
The paper studies Gauss maps of space-like stationary surfaces in Lorentz-Minkowski space, focusing on ramification and unicity.
problem Value distribution properties of Gauss maps on space-like stationary surfaces.
method Investigation of ramification and unicity properties, considering rational graphic Gauss images.
result Obtained general conclusions similar to Euclidean space, extending to rational graphic Gauss images.
New examples of mixed-type zero-curvature graphs found.
problem Finding new examples of zero-curvature graphs in Lorentz-Minkowski space.
method Using Konderak's representation formula to construct entire zero-curvature graphs over specific planes.
result Existence of new types of entire zero-curvature graphs in mixed-type in Lorentz-Minkowski space.
Extends Newton's minimal resistance problem to Lorentz-Minkowski space.
problem Minimal resistance in Lorentz-Minkowski space.
method Derived functional energy, determined Euler-Lagrange equation, analyzed maximum principle, found separable and radial solutions.
result Obtained solutions with conical singularities at the origin and analyzed the Single Shock Condition.
We study the third Laplace-Beltrami operator of timelike rotational surface of (T,L)-type in three dimensional Lorentz-Minkowski space.
Classifies zero mean curvature surfaces in Lorentz-Minkowski space.
problem Classifying surfaces with zero mean curvature.
method Using separable surface equations and constructing examples.
result All zero mean curvature surfaces of separable type have been classified.
Topics in Lorentz Geometrymath.DG
Lecture notes from the mini-course "Topics in Lorentz Geometry" taught at the University of São Paulo, in March/2019. The text has three parts: (i) an overall view of linear algebra in the pseudo-Euclidean space , with focus on Lorentz-Minkowski space and its role in Physics; (ii) a version of the Funda…
We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the …
The paper classifies maximal translation surfaces in Lorentz-Minkowski space.
problem Classifying maximal translation surfaces in Lorentz-Minkowski space.
method Analyzing surfaces defined as the sum of two spatial curves, proving properties of generating curves, and classifying surfaces based on curve types.
result A full description of maximal translation surfaces, including new examples not found in Euclidean space.
The paper classifies ruled surfaces in a specific space that move in a special way.
problem Understanding how surfaces move in a special space.
method Classifying ruled surfaces as translating solitons of inverse mean curvature flow.
result Existence of non-cylindrical ruled translating solitons.
The paper classifies ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
problem Classifying ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
method Maximum principle applications, classification based on causal character of rulings.
result Planes are the only cylindrical stationary surfaces. For non-cylindrical surfaces, classification depends on the causal character of the rulings.
Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.
problem Analyzing mixed types of curves with singular points in Lorentz-Minkowski 3-space.
method Using lightcone frame to consider Bertrand types for lightcone framed curves.
result Existence conditions of Bertrand lightcone framed curves in all cases.
We investigate a variational problem in the Lorentz-Minkowski space whose critical points are spacelike surfaces with constant mean curvature and making constant contact angle with a given support surface along its common boundary. We show that if the support surface is a pseudosphere, then the surface is a plana…
We review part of the classical theory of curves and surfaces in -dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.
We construct triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space, with the same topology as the triply periodic minimal surfaces in the Euclidean 3-space, called Schwarz rPD surfaces.
Embedded triply periodic zero mean curvature surfaces of mixed type in Lorentz-Minkowski 3-spacemath.DG
We construct embedded triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space with the same topology as the Schwarz D surface in the Euclidean 3-space.
In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper's surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space L3.
Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.
problem Understanding the curvature properties of modular surfaces in Lorentz-Minkowski space.
method Analyzing the sign of Gaussian and mean curvature, classifying surfaces, and applying to conformal field theories.
result Complete classification of zero Gaussian curvature modular surfaces and non-existence of non-planar maximal modular surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.
problem Investigate differential geometric properties of lightcone framed surfaces.
method Introduced modified frame to study the properties of lightcone framed surfaces.
result Showed behavior of Gaussian and mean curvatures at lightlike and singular points.
Extends Euler's problem to Lorentz-Minkowski plane.
problem Finding critical points of moment of inertia in Lorentz-Minkowski space.
method Explicit solutions for stationary curves, symmetries, inversions, and energy maximization.
result Explicit solutions for stationary spacelike and timelike curves, and methods to transform between them.
We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.
In 3-dimensional Lorentz-Minkowski space we determine the number of catenoids connecting two coaxial circles in parallel planes. This study is separated according to the types of circles and the causal character (spacelike and timelike) of the catenoid.
Study the geometry of lightlike loci on mixed type surfaces in Lorentz-Minkowski 3-space.
problem Characterize the differential geometric properties of lightlike loci on mixed type surfaces.
method Define a frame field and lightlike ruled surfaces along the lightlike locus, analyze their singularities and intersections.
result Establish a relationship between the singularities of lightlike ruled surfaces and the differential geometric properties of the lightlike locus.
Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in Minkowski 3-spa…
We introduce a new approach to the study of timelike minimal surfaces in the Lorentz-Minkowski space through a split-complex representation formula for this kind of surface. As applications, we solve the Björling problem for timelike surfaces and obtain interesting examples and related results. Using the Björling repre…
Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space and maximal surfaces in Lorentz-Minko…
The paper estimates curvature for specific hypersurfaces in a special space.
problem Estimating curvature for spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Analyzing the geometry of spacelike admissible graphic hypersurfaces.
result Existence of specific hypersurfaces with prescribed curvature and boundary conditions.
The paper studies how spacelike surfaces evolve in Lorentz-Minkowski space over time.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Inverse Gauss curvature flow with Neumann boundary condition.
result The evolving surfaces converge to a constant function as time goes to infinity.
The paper studies umbilics on surfaces in Lorentz-Minkowski space.
problem Properties of umbilics on space-like or time-like surfaces in .
method Analyzes curvature line flows and proves properties of umbilics.
result Existence of germs with isolated umbilics of various indices.
For a regular curve on a spacelike surface in Lorentz-Minkowski -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.
Pogorelov type estimates for a class of Hessian quotient equations in Lorentz-Minkowski space math.AP
The paper establishes Pogorelov type estimates for solutions to Hessian quotient equations in hyperbolic space.
problem Estimating solutions to Hessian quotient equations in Lorentz-Minkowski space.
method Using a priori estimates, the paper establishes Pogorelov type estimates for k-convex solutions.
result Pogorelov type estimates for k-convex solutions to Hessian quotient equations in hyperbolic space.
The geometry and topology of complete nonorientable maximal surfaces with lightlike singularities in the Lorentz-Minkowski 3-space are studied. Some topological congruence formulae for surfaces of this kind are obtained. As a consequence, some existence and uniqueness results for maximal Moebius strips and maximal Klei…
The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces formed by pseudo null, partially null, and null curves.
method Obtained parametric expressions and geometric invariants of canal hypersurfaces.
result Characterizations of tubular hypersurfaces in .
The paper studies a flow of spacelike surfaces in Lorentz-Minkowski space, proving convergence to a hyperbolic plane.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Anisotropic inverse mean curvature flow with Neumann boundary condition.
result The flow converges to a hyperbolic plane as time tends to infinity.