arXiv research

The paper extends Hawking--Page solutions to various spacetimes with singularities.

Proves inextendibility of weak null singularities from curvature blow-up.

New black hole models with both null and spacelike singularities.

The paper proves a conjecture about spacetimes and singularities.

Study on unique spacetime extensions in 1+1 dimensions with applications to weak null singularities.

The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.

Simplified proof of cosmic singularity theorem using new mathematical techniques.

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

Study shows instability of naked singularities in scalar field models.

Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.

This is a survey article on recognition problem of frontal singularities. We specify geometrically several frontal singularities and then we solve the recognition problem of such singularities, giving explicit normal forms. We combine the recognition results by K. Saji and several arguments on openings, which was perfo…

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.

We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant. More specifically, for 3+1 dimensional spacetimes which satisfy the null energy condition and contain a future expanding compact Cauchy surface, we establish a precise c…

New perspective on Ricci flow on spheres using Minkowski spacetime.

We consider a class of globally hyperbolic space-times with "expanding singularities". Under suitable assumptions we show that no $C^0$-extensions across a compact boundary exist, while the boundary must be null wherever differentiable (which is almost everywhere) in the non-compact case.

The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.

Synthetic framework for null hypersurfaces in non-smooth spacetimes.

The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.

In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simp…

Cosmological singularity theorems such as that of Hawking and Penrose assume local curvature conditions as well as global ones like the existence of a compact (achronal) slice. Here, we prove a new singularity theorem for chronological spacetimes that satisfy what we call a `past null focusing' condition. Such a condit…

New null distance bounds confirm Big Bang singularity in cosmological models.

Study of maximal surfaces in a specific Heisenberg group with singularities.

The paper strengthens a singularity theorem in General Relativity.

Study shows singular sets for certain fluid equations are negligible.

The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.

We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ paralle…

Constructs Lorentzian harmonic maps and associated timelike surfaces.

In the Compressed Sensing community, it is well known that given a matrix $X \in \mathbb R^{n\times p}$ with $\ell_2$ normalized columns, the Restricted Isometry Property (RIP) implies the Null Space Property (NSP). It is also well known that a small Coherence $μ$ implies a weak RIP, i.e. the singular values of $X_T$ l…

New theorem shows black holes are real, with clearer event horizon.

Study null curves and their motion in 3D flat space-time, leading to integrable hierarchies.

In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points o…

We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, $\mathbb{L}^3$, using the identification of $\mathbb L^3$ with $\mathbb C\times \mathbb R$. Further we give a different proof for the singular Björling problem for the case of closed real analytic null curve. As an application, …

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the sing…

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null…

New Weyl's laws discovered for compact spaces with Ricci curvature bounds.

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

Minimal surfaces and curves can have singularities removed by isotopy.

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

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and…

The paper extends Nakamaye's theorem to non-closed forms on complex manifolds.

We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singu…

It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner-Nordström data for the Einstein-Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continu…

Study on overlaps of singular vectors in Gaussian matrix submatrices.

Study bounds singular set of minimal hypersurfaces with index control.

We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine--Tristram signatures associated with it, defined for null-homologous…