arXiv research

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …

New results on non-existence and rigidity of spacelike submanifolds in spacetimes.

In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvatur…

New method discovers mean and variance causal graphs from heteroscedastic data.

This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they a…

Study on surfaces in Heisenberg group with constant mean curvature.

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found …

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space I_{p}^{3} that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron in order for spacelike and timelike curves. The causal character of all admissible surfaces in…

With several concrete examples of zero mean curvature surfaces in $\boldsymbol{R}^3_1$ containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first …

New curvature measure for causal sets derived from optimal transport.

The paper establishes curvature inequalities and rigidity results for CMC and STCMC surfaces in Riemannian and Lorentzian geometry.

The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic fu…

Detect spacetime curvature without rulers and clocks in 3D.

New method recovers causal order from dependent data.

We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with $C^{1,1}$-metrics with a lower bound on the tim…

In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize "ideal cones" (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.

TS-K-means improves financial data clustering with dynamic time warping.

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof o…

New CMC existence result for expanding cosmological spacetimes.

Causal spacetimes with Ricci tensor have unique transformations.

In this work, we study spacelike and timelike surfaces of revolution in Minkowski space $\e_{1}^{3}$ that satisfy $aH+bK=c$, where $H$ and $K$ denote the mean curvature and the Gauss curvature of the surface and $a$, $b$ and $c$ are constants. The classification depends on the causal character of the axis of revolution…

Study invariant $λ$-translators in Lorentz-Minkowski space.

A new metric compares true and learned causal graphs considering data and graph structure.

Detect spacetime curvature with event causality measurements.

Example of spacetime with causal bubbling, splitting into timelike and spacelike parts.

Unified approach for estimating quantiles of potential outcomes using inverse estimating equations.

Pattern recognition in neuroimaging distinguishes between two types of models: encoding- and decoding models. This distinction is based on the insight that brain state features, that are found to be relevant in an experimental paradigm, carry a different meaning in encoding- than in decoding models. In this paper, we a…

New relation between curvature bounds and spacetime inextendibility.

New relation between curvature bounds and spacetime inextendibility.

Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.

New tests for distributional causal effects using improved kernel estimators.

The paper tackles matching a desired mean in causal systems through shift interventions.

BayesIMP combines multiple causal graphs to estimate average treatment effects with uncertainty.

This paper looks at the splitting problem for globally hyperbolic spacetimes with timelike Ricci curvature bounded below containing a (spacelike, acausal, future causally complete) hypersurface with mean curvature bounded from above. For such spacetimes we show a splitting theorem under the assumption of either the exi…

Proposes a method to evaluate generalizability in causal inference models.

Causal inference from observational data is hard due to discontinuous causal effects.

New methods for estimating complex causal effects in econometrics.

This article is concerned with causal structures, which are defined as a field of tangentially non-degenerate projective hypersurfaces in the projectivized tangent bundle of a manifold. The local equivalence problem of causal structures on manifolds of dimension at least four is solved using Cartan's method of equivale…

Study on trapped submanifolds in spacetimes, proving rigidity and non-existence results.

Proposes Causal k-Means Clustering to identify subgroup effects.

The paper extends Hawking's singularity theorem to metrics with Hölder continuity and bounded curvature.

New model improves data augmentation for causal tasks.

The concept of closed trapped surface is of paramount importance in General Relativity and other gravitational theories. However, it is a purely geometrical object. With the aim of bringing this concept to closer attention by the mathematical community, I introduce the generalized idea of trapped submanifold by using t…

Proposes a new model to identify unknown counterfactual outcomes for continuous variables.

One of the central difficulties of settling the $L^2$-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of En…

Defines tangent spaces on causal sets using partial derivatives and metrics.

This paper introduces a new method to better understand financial market causality.