Paper proves isoperimetric inequality for Minkowski spacetime.
problem Maximizing volume of domain of dependence for finite lightcone.
method Lorentz polarisation to study variational problem.
result Maximal volume achieved by spacelike hyperplane truncated finite lightcone.
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.
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.
We define the notions of St1×Ss1-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…
Past lightcones of certain points in globally hyperbolic spacetimes determine the entire spacetime.
problem Determining the entire spacetime from the past lightcone of a point.
method Analyzing properties of globally hyperbolic spacetimes and using null lines and observer horizons.
result Past lightcones of certain points in globally hyperbolic spacetimes determine the entire spacetime (up to isometry).
Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.
problem Analyzing singularity formation and asymptotic behavior of null mean curvature flow.
method Rescaling procedure to relate to 2d-Ricci flow, singularity analysis, asymptotic behavior study.
result Ancient solutions to the flow can be understood in terms of 2d-Ricci flow.
Proves estimate similar to De Lellis-Müller on Minkowski lightcone.
problem Estimating spacelike cross sections of the Minkowski lightcone.
method Geometric scaling invariant estimate, using singularity models and almost-Schur lemma.
result Spacelike cross sections are W2,2-close to a round surface. 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.
problem Creating foliations of lightcones with specific geometric properties.
method Employing a geometric flow inspired by Huisken-Yau's approach for Riemannian settings.
result Initial data converges exponentially to an STCMC surface under area preserving null mean curvature flow.
Paper proves uniqueness of specific spacetime surfaces in a lightcone.
problem Proving uniqueness of surfaces of constant spacetime mean curvature in a lightcone.
method Used a fairly generic notion of asymptotic flatness to prove uniqueness.
result Unique foliation by surfaces of constant spacetime mean curvature exists under weaker assumptions.
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.
New perspective on Ricci flow on spheres using Minkowski spacetime.
problem Classifying singularity models for null mean curvature flow in Minkowski spacetime.
method Equivalence of 2d-Ricci flow and null mean curvature flow on lightcones.
result Classification of singularity models for null mean curvature flow.
Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
Study on stability of surfaces in null cones under area-preserving variations.
problem Investigating stability of spacelike cross sections of null cones.
method Area-preserving variations, Hawking energy analysis, spherical cross sections.
result Only round spheres are stable cross sections of the standard Minkowski lightcone.
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≥4. After generalising three distinct quasilocal energy definitions to higher dimensions under appropriate assumptions, we evaluate their small sp…
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.
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 …
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.
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…
We prove effective uniformization for nearly round 2-spheres and investigate their stability.
problem Proving effective uniformization for nearly round 2-spheres and their stability.
method Utilizing an identity related to the third-order differential of the conformal factor, and an isometric embedding of a round sphere into Euclidean space using an orthogonal basis of the first eigenspace of the Laplacian operator.
result We provide a simplified proof of effective uniformization and its stability.
Generalizes Fermat's principle for wave propagation in cone structures.
problem Wave propagation in complex media with discontinuities and anisotropy.
method Generalizes Fermat's principle to smooth interfaces separating two cone structures representing wave propagation in various media.
result Conditions for critical points of arrival time functional, generalizing Snell's law and reflection.
To study spacelike surfaces in the Lorentz-Minkowski space R14, we construct a pair of maps whose values are in the lightcone, called lr±-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…
In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points …
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.
Gauging procedure constructs lagrangians for carrollian gravity.
problem Constructing lagrangians for carrollian gravity.
method Gauging procedure applied to Klein pairs corresponding to homogeneous spaces.
result Generalizes first-order lagrangians for four-dimensional maximally symmetric carrollian spaces.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
Study of spacelike discs in Minkowski cones, proving self-similar expansion.
problem Mean curvature flow of spacelike discs in Minkowski cones.
method Analysis of parabolic boundary value problem for self-similar solutions.
result Existence of solutions rescaling to self-similarly expanding solutions.
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…
Sharp Sobolev inequality on circle proven with Lorentz invariance.
problem Proving a sharp Sobolev inequality on the circle.
method Rearrangement inequality, variational method, Lorentz transformation.
result Sharp inequality and global minimizer on circle.
In General Relativity the metric can be recovered from the structure of the lightcones and a measure giving the volume element. Since the causal structure seems to be simpler than the Lorentzian manifold structure, this suggests that it is more fundamental. But there are cases when seemingly healthy causal structure an…
New spaces at infinity identified for Minkowski spacetime.
problem Characterizing asymptotic infinities of Minkowski spacetime.
method Embedding and describing homogeneous spaces of the Poincaré group.
result Determined new structures on asymptotic infinities.
The study characterizes surfaces with specific harmonic properties in pseudo-conformal geometry.
problem Characterizing surfaces with harmonic properties in pseudo-conformal geometry.
method Investigating sphere congruences, quasi-umbilical surfaces, and constant mean curvature surfaces.
result Generically, Bryant's quartic differential is divergence free if and only if the surface is superconformal or orthogonal to a harmonic congruence of spheres.
Compactifies Minkowski space using unitary matrices.
problem Compactifying Minkowski space for quantum field theories.
method Using Cayley transform and unitary group $\U(2)$.
result Defines interesting backgrounds for quantum field theories.
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…
This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime R1+3. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime R1+3, the …
21cmEMU speeds up EoR simulations by 10^4x, predicting key observables with sub-percent accuracy.
problem Efficiently simulating and inferring Epoch of Reionization properties from limited computational resources.
method Developed an emulator for 21cmFAST simulation outputs, reducing computational cost by 10^4x.
result Sub-percent accurate predictions of 6 key observables from EoR simulations.
The paper proves rigidity theorems for space-like hypersurfaces in Minkowski space.
problem Stability and rigidity of space-like hypersurfaces in Minkowski space.
method Weinberger-type approach with P-functions and integral identities.
result Hypersurfaces with constant higher-order mean curvature ratios and specific boundary conditions are parts of hyperboloids.
Finite vector bundles over complex manifolds are trivializable via finite covers.
problem Understanding when holomorphic vector bundles over compact complex manifolds are trivializable.
method Introducing finite bundles and using finite étale covers to trivialize holomorphic vector bundles.
result Holomorphic vector bundles over compact complex manifolds are finite if and only if they admit a flat holomorphic connection with finite monodromy.
Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.
problem Existence of non-residually finite hyperbolic groups
method Direct implication
result Existence of non-residually finite rigid hyperbolic groups
Finite type and finitely generated homotopy groups for manifold automorphisms.
problem Finite type and homotopy group properties of manifold automorphism spaces.
method Analyzing the classifying space of diffeomorphism groups and using simple homotopy theory.
result The classifying space of diffeomorphism groups has finitely generated homotopy groups.
The study proves how groups can be split with limited complexity.
problem Understanding the complexity of group splittings.
method Analyzing trees with finite stabilizers and their quotient structures.
result Deformation spaces of trees have maximal complexity.
Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these resul…
In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…
Study of uncountable family of finitely generated groups.
problem Characterize a family of finitely generated residually finite groups.
method Examined groups of the form F2∗HF2 where F2 is a rank-2 free group and H is an infinitely generated subgroup. result Unveiled uncountable family of groups up to isomorphism.
Study on minimal submanifolds with finite curvature in Euclidean space.
problem Finite diffeomorphism types of complete immersed minimal submanifolds with finite total curvature.
method Adapted method from Chodosh, Ketover, and Maximo for hypersurfaces to submanifolds of arbitrary codimension.
result Proved finite diffeomorphism types for complete immersed minimal submanifolds with finite total curvature.