arXiv research

Solves surface problem in 3D light cone.

Classifies surfaces with zero mean curvature in a light cone.

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

Curvature flow and inverse curvature flow solutions on 2D light cone identified.

New inequality shows all special submanifolds in light cone are totally umbilical spheres.

The paper studies volumes of conformally flat manifolds in light-cone geometry.

In this paper, we show the rigidity of isometric immersions for a Riemannian manifold of dimension $n-1$ into the light cone of $n+1$ dimensional Minkowski, de Sitter and anti-de Sitter spacetimes for $n\geq 3$.

We construct stationary flat three-dimensional Lorentzian manifolds with singularities that are obtained from Euclidean surfaces with cone singularities and closed one-forms on these surfaces. In the application to (2+1)-gravity, these spacetimes correspond to models containing massive particles with spin. We analyse t…

We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of these cones. The affine spheres are represented by explicit hypersurface immersions …

The paper classifies periodic solitons in curve flows on the light-cone.

Spatio-temporal data is intrinsically high dimensional, so unsupervised modeling is only feasible if we can exploit structure in the process. When the dynamics are local in both space and time, this structure can be exploited by splitting the global field into many lower-dimensional "light cones". We review light cone …

On a time-oriented Lorentzian manifold $(M,g)$ with non-empty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets $S\subset M$ of sources is uniquely determined by measurements of the intersection of future light cones from points in $S$ …

In this paper we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian $\mathbb{R}^4$, invariants under the centro-affine action of the Lorentzian group, is Poisson equivalent to the space of conformal differential invariants for curves in the Möbius sphere…

In the Batalin-Vilkovisky formalism, gauge conditions are expressed as Lagrangian submanifolds in the space of fields and antifields. We discuss a way of patching together gauge conditions over different parts of the space of fields, and apply this method to extend the light-cone gauge for the superparticle to a conic …

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically fini…

Measuring supernova neutrinos removes spacetime's conformal freedom.

Study path geometries with constant torsion and cone structures.

The paper classifies orbits of $SO(3,1)$ in a 4D Minkowski space.

Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.

Let O be a three-dimensional Nil-orbifold, with branching locus a knot Sigma transverse to the Seifert fibration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in (pi-epsilon, pi). We also study the space of Dehn filling parameters of O-Sigma. Surprisingly it is not diffeomorphic to the defo…

Study connects contact structures to cone geodesics and contactomorphisms.

We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper. As an application we show that there is an open set in the space of cones in the …

We prove a number of results relating various measures (volume, Legendrian index, stability index, and spectral curve genus) of the geometric complexity of special Lagrangian $T^2$-cones. We explain how these results fit into a program to understand the "most common" three-dimensional isolated singularities of special …

New cones in 4D space found with minimal mass.

This paper mainly aims to establish the well-posedness on time interval $[0,\varepsilon^{-\frac{1}{2}}T]$ of the classical initial problem for the bosonic membrane in the light cone gauge. Here $\varepsilon$ is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifo…

Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.

Study of spacelike discs in Minkowski cones, proving self-similar expansion.

We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of $C^\infty$ norms on $\R^3$ admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math \textbf{166}, 55--83), and gives the exist…

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrighta…

The moduli space of Riemann surfaces with at least two punctures can be decomposed into a cell complex by using a particular family of ribbon graphs called Nakamura graphs. We distinguish the moduli space with all punctures labelled from that with a single labelled puncture. In both cases, we describe a cell decomposit…

We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study hyperbolic structures on three-dimensional manifolds

The paper classifies submanifolds in a specific Lorentz-Minkowski space.

Study on extremals in sub-Lorentzian geometry defined by antinorm.

The common assertion that the Ricci flows of Einstein spaces with cosmological constant can be modelled by certain classes of nonholonomic frame, metric and linear connection deformations resulting in nonhomogeneous Einstein spaces is examined in the light of the role played by topological three dimensional (3D) Taub-N…

For every proper convex cone $K \subset \mathbb R^3$ there exists a unique complete hyperbolic affine 2-sphere with mean curvature $-1$ which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This eq…

We solve the analogue of Björling's problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve $y_0$ in $S^3$, together with the prescription of the values of the surface normal and the dual Willmore surface along the c…

Two single parameter families of polyhedra $P(ψ)$ are constructed in three dimensional spaces of constant curvature $C(ψ)$. Identification of the faces of the polyhedra via isometries results in cone manifolds $M(ψ)$ which are topologically $S^1\timesS^2$, $S^3$ or singular $S^2$. The singular set of $M(ψ)$ can have se…

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a comp…

In the present paper, we consider a position vector of an arbitrary curve in the three-dimensional Galilean 3-space. Furthermore, we give some conditions on the curvatures of this arbitrary curve to study special curves and their Smarandache curves. Finally, in the light of this study, some related examples of these cu…

In this work we simulate null geodesics for the Bonnor massive dipole metric by implementing a symbolic-numerical algorithm in Sage and Python. This program is also capable of visualizing in 3D, in principle, the geodesics for any given metric. Geodesics are launched from a common point, collectively forming a cone of …

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path i…

Researchers extend parametrization of Margulis spacetimes using strip deformations.

We construct and study a family of double-periodic almost entire solutions of the maximal surface equation. The solutions are parameterized by a submanifold of $3\times 3$-matrices (the so-called generating matrices). We show that the constructed solutions are either space-like or of mixed type with the light-cone type…

New mappings in Minkowski spacetime classified under mild conditions.

Introduces a new Yang-Mills functional for connections and scalars over circle bundles.

Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.

A general theory of rigid completely integrable analytic partial differential equations is endeavoured. The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal dimension equal to 10. Explicit formulas for the Lie prolongation of vector fields t…

We construct a cut-off version of nonpertubative closed Bosonic string field theory in the light-cone gauge with imaginary string coupling constant. We show that the partition function is a continuous function of the string coupling constant, and conjecture a relation between the formal power series expansion of this p…