arXiv research

Future stability of FLRW solutions in expanding 3D space is shown for compact perturbations.

Study on future stability of FLRW spacetime solutions with decelerated expansion.

The paper studies the past inextendibility of FLRW spacetimes using the VDR asymptote.

New proof shows FLRW spacetimes can't be extended smoothly in certain axisymmetric cases.

Proves some flat spacetimes can't be extended smoothly.

The paper simplifies FLRW photon propagators using geometric embeddings.

New approach classifies conformal Killing vector fields for FLRW space-time.

In the present paper we provide new examples of marginally trapped surfaces and tubes in FLRW spacetimes by using a basic relation between these objects and CMC surfaces in 3-manifolds. We also provide a new method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on the classical Hopf …

The paper proves that certain FLRW spacetimes cannot be extended past the big bang.

The existence, established over the past number of years and supporting earlier work of Ori [14], of physically relevant black hole spacetimes that admit $C^0$ metric extensions beyond the future Cauchy horizon, while being $C^2$-inextendible, has focused attention on fundamental issues concerning the strong cosmic cen…

Based on a general $(1+3)$ threading of the spacetime $(M,g)$, we obtain a new and simple splitting of a both the Einstein field equations (EFE) and the conservation laws in $(M,g)$. As an application we obtain the splitting of (EFE) in an almost FLRW universe with energy-momentum tensor of a perfect fluid. In particul…

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric $g_0$ consists of a set of tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$, its Riemann curvature and their derivatives, that are satisfied if and only if $g$ is loc…

The paper examines gravitational singularities in spacetimes and proves inextendibility.

The study explores spacetimes with changing spatial curvature, leading to topological transitions.

Exploring distance functions on spacetime models.

The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity. In previous papers it has been proposed an equivalent form of Einstein's equation, which extends i…

The paper examines isotropic cosmological space-times with changing sectional curvature.

This work presents the foundations of Singular Semi-Riemannian Geometry and Singular General Relativity, based on the author's research. An extension of differential geometry and of Einstein's equation to singularities is reported. Singularities of the form studied here allow a smooth extension of the Einstein field eq…

The paper proves stability of certain cosmological models with negative spatial curvature.

New cosmological spacetimes without CMC Cauchy surfaces found.

New findings show cosmological constant as initial condition for non-isotropic spacetimes.

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, w…

We study generalizations of Lorentzian warped products with one-dimensional base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. G…

Milne-like spacetimes are a class of FLRW models which admit $C^0$ spacetime extensions through the big bang. The boundary of a Milne-like spacetime can be identified with a null cone in the extension. We find that the comoving observers all emanate from a single point in the extension. This suggests that something phy…

New cosmological models with changing curvature slices.

A series of old and recent theoretical observations suggests that the quantization of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high energy scales. But an identification of the deep mechanism causing this dimens…

The paper establishes criteria for spacetime inextendibility using asymptotic volume-distance-ratio analysis.

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface $Σ$ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field $α$-att…

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

Unique ancient solutions found for anisotropic curve shortening flow.

The paper constructs solutions to a critical Dirac equation on spheres.

Paper classifies ancient solutions to 3D Ricci flow.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Study higher-dimensional Ricci flow solutions, proving uniqueness.

Ancient solutions of Ricci flow with Type I growth are classified.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.

New ancient solutions found for curvature flow in 2D.

Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty,T)$ for some $T\in\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient so…

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.

Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

Generic level sets in mean curvature flow are BV solutions.

Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.

In this paper we study the difference between algebraic and geometric solutions of the hyperbolic Dehn filling equations for ideally triangulated 3-manifolds. We show that any geometric solution is an algebraic one, and we prove the uniqueness of the geometric solutions. Then we do explicit calculations for three inter…

Novikov equation symmetries, solutions, and pseudo-spherical surfaces studied.