No global solutions found for time-like minimal submanifolds in Minkowski space.
problem Existence of global-in-time axisymmetric solutions to time-like minimal submanifolds in Minkowski space.
method Analysis of limiting geometry as maximal time of existence is approached.
result No global solutions found for time-like minimal submanifolds in Minkowski space.
Study minimal time-like surfaces in 4D space-time, proving conditions for existence.
problem Characterize minimal time-like surfaces in 4D space-time.
method Apply complex analysis over double numbers to classify surfaces and derive natural equations.
result Existence and uniqueness of minimal time-like surfaces based on curvature conditions.
Canonical coordinates defined for minimal time-like surfaces in n-dimensional Minkowski space.
problem Characterizing canonical coordinates on minimal time-like surfaces.
method Introducing canonical coordinates and proving their existence and uniqueness; using analysis over the algebra of double numbers.
result Canonical coordinates on minimal time-like surfaces are characterized by a natural condition for a complex function over the algebra of double numbers.
Study Born-Infeld solitons and solve Björling problem for them.
problem Existence and non-uniqueness of solutions to the Björling problem for Born-Infeld solitons.
method Two approaches: treating as time-like minimal surfaces or using Barbashov-Chernikov representation.
result Solution to Björling problem may not be unique.
The paper studies sections of time-like twistor spaces with specific covariant derivatives.
problem Sections of time-like twistor spaces with light-like or zero covariant derivatives.
method Analyzes conformal Gauss maps of time-like minimal surfaces and properties of almost paracomplex structures.
result Sections of time-like twistor spaces have light-like or zero covariant derivatives.
Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…
Two self-similar solutions found for time-like hypersurfaces in Minkowski spacetime.
problem Finding self-similar solutions for time-like extremal hypersurfaces in Minkowski spacetime.
method Explicit construction of two self-similar solutions.
result An untable eigenvalue found in the linearized equation around the solutions.
Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space S12 and introducing space-like height function on the unit speed time-like curves on S12, the invariants of the unit speed time-like curves on S12 and geometric properties of de Si…
We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface we introduce locally natural principal parameters and prove that such a surface is determined uniquely (up to motion) by a special invariant function, which satisfies a natural no…
Study links Hopf differentials to curvature line flows on time-like CMC surfaces.
problem Understanding the relationship between Hopf differentials and curvature line flows on time-like CMC surfaces.
method Investigation of Hopf differentials and curvature line flows on time-like CMC surfaces in Lorentzian 3-space forms.
result The index of a curvature line flow at an umbilic point depends on the remainder of the Hopf differential's order modulo four.
Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.
problem Characterize space-like and time-like surfaces in Robertson-Walker space-times with positive relative nullity.
method Provide necessary and sufficient conditions, local classification theorems, and analyze special spaces.
result Local classification theorems for space-like and time-like surfaces in L14(f,0) with positive relative nullity. Study variational problem for time-like curves in Einstein universe.
problem Variational problem for time-like curves in Einstein universe.
method Conformally invariant variational problem, analysis of stationary curves, integration by quadratures.
result Stationary curves are trapped into Einsetin universes of dimension 2, 3, or 4.
Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…
We study the neutral Kähler metric on the space of time-like lines in Lorentzian E13, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metr…
In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E13 are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E13 satisfies a vector differential equation of fourth order. The general so…
Study of generalized Bishop frames on time-like curves in 4D Lorentz space.
problem Characterize frames for time-like curves in 4D Lorentz space.
method Introduced and studied generalized Bishop frames for regular time-like curves in 4D Lorentz space.
result Hierarchy of frames exists for time-like curves in 4D Lorentz space, similar to Euclidean case.
In a 2004 paper, Lindblad demonstrated that the minimal surface equation on Rl1,1 describing graphical time-like minimal surfaces embedded in R1,2 enjoy small data global existence for compactly supported initial data, using Christodoulou's conformal method. Here we give a different, geometr…
The study defines invariants for time-like surfaces with real asymptotic lines.
problem Characterizing time-like surfaces with real asymptotic lines.
method Fundamental theorem of Bonnet-type, canonical parameters, invariant functions, PDEs.
result Time-like surfaces are determined by four invariant functions, two of which can be Gauss and mean curvature.
The paper classifies time-like surfaces in a static space-time.
problem Classifying time-like surfaces in a static space-time.
method Constructing a pseudo-orthonormal frame field and analyzing invariants.
result Complete classification theorem for class~A surfaces. Study on surfaces in neutral space forms with zero mean curvature.
problem Characterizing surfaces with zero mean curvature in neutral space forms.
method Analyzing curvature and normal connection properties of time-like conformal immersions.
result Conditions for surfaces with zero mean curvature in neutral space forms.
The Jorge-Meeks n-noid (n≥2) is a complete minimal surface of genus zero with n catenoidal ends in the Euclidean 3-space R3, which has (2π/n)-rotation symmetry with respect to its axis. In this paper, we show that the corresponding maximal surface fn in Lorentz-Minkowski 3-space $\boldsymb…
Paper proves a new criterion for time-like geodesics in flat spacetimes.
problem Existence and nature of time-like geodesics in asymptotically flat spacetimes.
method Generalized topological criterion using the Jordan-Brouwer Separation Theorem and differential geometry.
result Conclusively affirms the presence of time-like geodesics intersecting transversally.
The paper studies umbilics on surfaces in Lorentz-Minkowski space.
problem Properties of umbilics on space-like or time-like surfaces in L3. method Analyzes curvature line flows and proves properties of umbilics.
result Existence of germs with isolated umbilics of various indices.
Study of closed trajectories in hyperbolic plane with specific curvature constraints.
problem Critical trajectories in hyperbolic plane for a specific energy function.
method Classification of critical trajectories based on momentum causal character, proof of existence of closed trajectories.
result Existence of countably many closed trajectories with time-like momentum.
Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
There are considered 4-dimensional pseudo-Riemannian spaces with inner products of signature (3,1) and (2,2). The objects of investigation are space-like and time-like hyperspheres in the respective cases. These hypersurfaces are equipped with almost contact B-metric structures. The constructed manifolds are characteri…
Uniqueness and stability of minimal submanifolds proved.
problem Uniqueness and stability of minimal submanifolds.
method Proved a strong stability condition on minimal submanifolds.
result Existence and convergence of mean curvature flow for minimal submanifolds.
Minimal Lagrangian submanifolds deform to J-minimal ones under small perturbations.
problem Understanding how minimal Lagrangian submanifolds behave under small perturbations in Kaehler-Einstein manifolds.
method Analyzing the deformation of minimal Lagrangian submanifolds under Kaehler-Einstein perturbations.
result Deformed submanifolds remain J-minimal under certain conditions.
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
Study shows area-minimizing submanifolds are mostly smooth except for specific types.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proved area-minimizing submanifolds are not generically smooth except for geodesics, minimal surfaces, and minimal hypersurfaces.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
Study shows area-minimizing submanifolds are mostly smooth except for geodesics, minimal surfaces, and hypersurfaces.
problem Understanding when area-minimizing submanifolds are smooth in mod 2 homology.
method Proving the mod 2 area-minimizing submanifolds are smooth in specific cases and establishing lower bounds on singular sets.
result Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces.
We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows mor…
Hasse principle applied to area-minimizing submanifolds across different homology types.
problem Understanding the behavior of area-minimizing submanifolds in various homology contexts.
method Extending the Hasse principle from number theory to geometric variational problems.
result Recovering information about area-minimizing submanifolds in integral homology from those in real and mod n homology. Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)-th Gauss-Bonnet curvature function, called (2k)-minimal su…
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
Uniqueness of minimal submanifolds in specific Riemannian manifolds.
problem Understanding uniqueness of minimal submanifolds in various Riemannian manifolds.
method Analyzing compact minimal submanifolds in specific classes of Riemannian manifolds.
result Uniqueness results for compact minimal submanifolds in large classes of Riemannian manifolds.
Study finds formulas for minimal submanifolds using Möbius transformations.
problem Understanding minimal submanifolds in Euclidean space.
method Monotonicity formulas for minimal submanifolds involving Möbius transformations.
result Proved formulas for minimal submanifolds under Möbius transformations.
Constructs minimal submanifolds in symmetric spaces using eigenfunctions.
problem Finding minimal submanifolds in symmetric spaces.
method Employing recent results from S. Gudmundsson and T.J. Munn, constructing submanifolds using eigenfunctions.
result Constructs minimal submanifolds of classical compact Riemannian symmetric spaces.
Paper studies second variation for L-minimal submanifolds in pseudo-Sasakian manifolds.
problem Analyzing stability of L-minimal submanifolds in pseudo-Sasakian manifolds.
method Provides a second variation formula and applies it to Lorentzian-Sasakian manifolds.
result Relates L-stability of Legendrians in a Sasakian manifold to their stability in an associated Lorentzian-Sasakian structure.
Develops methods for computing conformal invariants of submanifolds.
problem Computing conformal invariants of submanifolds.
method Direct construction of extrinsic ambient space, global invariants of conformally compact minimal submanifolds, introduction of conformal submanifold scalars.
result Derives an explicit Gauss--Bonnet--Chern-type formula and proves a rigidity result.
Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.
Minimal submanifolds in spheres can be produced via Clifford type minimal products, and their Morse indices and nullities are calculated.
problem Understanding the properties of minimal submanifolds in spheres via Clifford products.
method Analyzing the first eigenfunctions and Morse indices of minimal products of minimal submanifolds.
result The Morse index and nullity of the minimal product are calculated and shown for specific cases.
The paper classifies product minimal Lagrangian submanifolds in complex space forms.
problem Understanding minimal Lagrangian submanifolds in complex space forms.
method Examining submanifolds as Riemannian products with constant sectional curvature.
result Complete classification of product minimal Lagrangian submanifolds.
The paper studies stability and instability of minimal submanifolds in complex Einstein spaces.
problem Stability and instability of minimal submanifolds in complex Einstein spaces.
method Computation of index and nullity, investigation of stability, and algorithm for higher eigenvalues.
result Criterion for instability of minimal submanifolds in some cases.
New minimal submanifolds in spheres share properties of the Clifford torus.
problem Finding new minimal surfaces in spheres.
method Analyzing properties of the Clifford torus and extending to other minimal submanifolds.
result More minimal submanifolds in spheres have helicoidal properties.
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.
Paper constructs new minimal submanifolds in spheres by spinning given ones.
problem Creating new minimal submanifolds in spheres from given ones.
method Spin given minimal submanifolds by a curve γ in a balanced way. result Generates spiral minimal products forming a two-dimensional family.
The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.
problem Understanding the relationship between metric and causal geometry in Lorentzian spaces.
method Constructing a Lorentzian length space with an orthogonal splitting on a product of an interval and a metric space, and using synthetic time-like Ricci curvature bounds.
result Established sufficient conditions for global hyperbolicity and formulated time-like Ricci curvature bounds without push-up and regularity assumptions.