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 geometric flows of spacelike curves in Lorentz-Minkowski plane and proves their long-term behavior.
problem Investigating geometric flows of spacelike curves in Lorentz-Minkowski plane.
method Examining the evolution of spacelike curves along prescribed geometric flows, including curve shortening and mean curvature flows.
result The geometric flows of spacelike curves in Lorentz-Minkowski plane exist for all time and converge to specific curves as time tends to infinity.
The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.
problem Classifying curves with constant curvature in dual affine and Lorentz-Minkowski planes.
method Investigation of invariants under equiaffine transformations and explicit equations for curves with constant curvature.
result Curves with constant curvature in dual affine and Lorentz-Minkowski planes are classified.
The paper studies a flow of spacelike curves in a Lorentz-Minkowski plane, showing convergence to a constant function.
problem Evolution of spacelike graphic curves in Lorentz-Minkowski plane.
method Anisotropic inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving curves converge to a constant function as time tends to infinity.
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.
Riemann zero mean curvature examples in the Lorentz-Minkowski space are surfaces with zero mean curvature foliated by circles contained in parallel planes. In contrast to the Euclidean case, this family of surfaces presents new and rich features because of the variety of types of circles. In this paper, we give a geome…
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.
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.
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.
Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space L3 which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space E3 and maximal surfaces in Lorentz-Minko…
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 how certain spacelike surfaces evolve over time in a specific space.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving surfaces converge to a hyperbolic plane as time goes to infinity.
It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space R13 is called of mixed type if it changes causal type from space-like to time-like. In R13, Osamu Kobayashi found …
For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and bou…
In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space ł3 and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in ł3 of constant mean c…
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.
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.
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 Rνn, with focus on Lorentz-Minkowski space and its role in Physics; (ii) a version of the Funda…
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.
Study proves conditions for translating solitons to be planar.
problem Conditions for translating solitons to be planar.
method Investigates translating solitons as sums and products of curves, extends results to Lorentz-Minkowski space.
result Translating solitons can be planar if one component is planar.
New minimal surfaces found using a modified metric connection.
problem Finding minimal surfaces in Euclidean 3-space.
method Used a special semi-symmetric metric connection instead of the Levi-Civita connection.
result Found non-trivial minimal surfaces other than planes.
A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …
We prove that a maximal surface in Lorentz-Minkowski space L3 can be extended analytically along its boundary if the boundary lies in a plane meeting the surface at a constant angle.
New surfaces in Lorentz-Minkowski space with constant mean curvature identified.
problem Identifying surfaces with constant mean curvature in Lorentz-Minkowski space.
method Using a specific coordinate system and properties of the Weingarten endomorphism, the mean curvature is shown to be constant under certain conditions.
result Constant mean curvature surfaces identified, including spacelike and timelike Enneper surfaces.
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…
We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space L3 with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the…
Consider a surface S immersed in the Lorentz-Minkowski 3-space R13. A complete light-like line in R13 is called an entire null line on the surface S in R13 if it lies on S and consists of only null points with respect to the induced metric. In this paper, we show th…
A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…
Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\math…
It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been…
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.
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.
In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew (S) and mean (H) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that H and S can be interpr…
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.
We classify (spacelike or timelike) surfaces of revolution with zero f-mean curvature in G2×R1, the Lorentz-Minkowski 3-space R13 endowed with the Gaussian-Euclidean density e−f(x,y,z)=2π1e−2x2+y2. It is proved that an f-maximal surface of revolution is either a …
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.
New inequality shows all special submanifolds in light cone are totally umbilical spheres.
problem Characterizing submanifolds with parallel mean curvature in Lorentz-Minkowski spacetime.
method Established an integral inequality and used it to derive a rigidity result.
result All compact submanifolds with parallel mean curvature in light cone are totally umbilical spheres.
We prove that maximal annuli in L3 bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli hav…
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.
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.
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.
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.
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.
The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.
problem Understanding null hypersurfaces in Lorentz-Minkowski space.
method Analyzing L-complete null hypersurfaces as wave fronts in Euclidean space. result Most null wave fronts can be realized as restrictions of certain L-complete null wave fronts. 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.
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.