arXiv research

Proves well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.

Cauchy used infinitesimals in differential geometry and integral geometry.

Paper introduces robust kernel ridge regression using Cauchy loss for handling various noise types.

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface …

Geometric approach to Dirac operator evolution on spacetimes.

Principal Component Analysis (PCA) has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Princ…

New cosmological spacetimes without CMC Cauchy surfaces found.

Proves conditions for Cauchy horizons in low-regularity spacetimes.

We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…

We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.

Study the metric geometry of Cauchy hypersurfaces in spacetimes.

New boundary conditions solve Cauchy problem for Dirac operators on spacetimes.

The zoology of singularities for Lorentzian manifold is slightly more complicated than for Riemannian manifolds. Our present work study Cauchy-compact globally hyperbolic singular flat spacetimes with extreme BTZ-like singular lines. We use the notion of BTZ-extension of a singular spacetime introduced in a previous pa…

Unique solutions found for diffusive martingale problems.

The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.

Lorentzian distances to Cauchy surfaces fail to be locally equi-Lipschitz.

Proves existence of Killing fields in smooth spacetimes with compact Cauchy horizons.

The paper generalizes the Cauchy-Schwarz-Bunyakovsky inequality and applies it to elasticity problems.

Investigates new $F$-structures and their Cauchy-Riemann properties.

In this paper, we prove the infinite dimensionality of some local and global cohomology groups on abstract Cauchy-Riemann manifolds.

We prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to …

We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be no…

Solves geometric Cauchy problem for submanifolds with constant rank.

We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study also a variant of the inverse problem of the Cauchy problem and prove that the …

New vacuum spacetimes without CMC Cauchy surfaces found.

Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at most linear growth. In this note, we give a condition on the volatility that is …

Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.

How many are linear connections with prescribed Ricci tensor? How many are statistical structures? The questions are answered in the analytic case by using the Cauchy-Kowalewski theorem.

Study shows existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems.

We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature $S^1 \times S^2$ Cauchy surface also contains a maximal Cauchy surface. Combining …

Given a globally hyperbolic spacetime $M$, we show the existence of a {\em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $\R \times S$.

Study well-posedness of Faraday tensor problem on specific spacetime manifolds.

We present new results concerning the solvability, of lack thereof, in the Cauchy problem for the debar operator, with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples.

We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in…

The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.

Study on spin-zero rest-mass fields using conformal geometric method.

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

We study the Cauchy data spaces of the strongly Callias-type operators using maximal domain on manifolds with non-compact boundary, with the aim of understanding the Atiyah-Patodi-Singer index and elliptic boundary value problems.

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

We construct the solution to the periodic Cauchy problem of the Schrödinger flow on the sphere. Such construction of solutions is formulated explicitly and therefore a geometric algorithm of solving this periodic Cauchy problem follows. Theoretical and experimental results will be discussed.

Study well-poses Dirac operator problem with APS boundary conditions.

The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.

Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.

New method constructs spacelike data leading to trapped surfaces.

The paper extends completeness notions to low-regularity spacetimes.

We initiate a series of works where we study the interior of dynamical rotating vacuum black holes without symmetry. In the present paper, we take up the problem starting from appropriate Cauchy data for the Einstein vacuum equations defined on a hypersurface already within the black hole interior, representing the exp…

Proves compact Cauchy horizons have constant surface gravity under null energy condition.

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…