arXiv research

Paper finds necessary condition for logarithmic Minkowski problem in higher dimensions.

Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.

Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.

We consider the timelike minimal surface problem in Minkowski spacetimes and show local and global existence of such surfaces having arbitrary dimension $\geq 2$ and arbitrary co-dimension, provided they are initially close to a flat plane.

Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.

New theory shows deep networks adapt to data's intrinsic dimensionality even when data isn't on a low-dimensional manifold.

Upper bound on singular set dimension for area-minimizing currents.

Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.

A particular, yet relevant, particular case of the Penrose inequality involves null shells propagating in the Minkowski spacetime. Despite previous claims in the literature, the validity of this inequality remains open. In this paper we rewrite this inequality in terms of the geometry of the surface obtained by interse…

We prove that no Brunn--Minkowski inequality from the Riemannian theories of curvature-dimension and optimal transportation can by satisfied by a strictly subRiemannian structure. Our proof relies on the same method as for the Heisenberg group together with new investigations by Agrachev, Barillari and Rizzi on ample n…

Study the limiting shape of solutions to the L_p-Minkowski problem as p approaches negative infinity.

This article proves that the zero locus of a $\mathbb{Z}/2$ harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.

We prove that any regular domain in Minkowski space is uniquely foliated by spacelike constant mean curvature (CMC) hypersurfaces. This completes the classification of entire spacelike CMC hypersurfaces in Minkowski space initiated by Choi and Treibergs. As an application, we prove that any entire surface of constant G…

Sharp Minkowski inequality for convex surfaces in curved spaces.

The paper proves new Minkowski inequalities for flows in warped spaces.

Stability of Minkowski space-time in higher dimensions proven for arbitrary small perturbations.

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

General $(α, β)$ norms are an important class of Minkowski norms which contains the original $(α, β)$ norms. In this note, by studying the behavior of the Darboux curves of the indicatrix, we give a characterization of 3-dimensional general $(α, β)$ norms. By studying the isoperimetric properties of the indicatrix, as …

Given a constant vector field $Z$ in Minkowski space, a timelike surface is said to have a canonical null direction with respect to $Z$ if the projection of $Z$ on the tangent space of the surface gives a lightlike vector field. In this paper we describe these surfaces in the ruled case. For example when the Minkowski …

In this paper, the radiation field is defined for solutions to Einstein vacuum equations which are close to Minkowski space-time with spacial dimension $n\geq 4$. The regularity properties and asymptotic behavior of those Einstein vacuum solutions are established at the same time. In particular, the map from Cauchy int…

By studying $L^p$-combinations of strongly isomorphic polytopes, we prove the equivalence of the $L^p$-Brunn-Minkowski inequality conjectured by Böröczky, Lutwak, Yang and Zhang to the local version of the inequality studied by Colesanti, Livshyts, and Marsiglietti and by Kolesnikov and Milman, settling a conjecture of…

The paper proves a Minkowski inequality on specific Riemannian manifolds.

New curvature-dimension condition for Lagrangians on manifolds.

The paper proves conjectures about Minkowski norms with specific symmetry groups.

We consider Finsler submanifolds $M^n$ of nonnegative Ricci curvature in a Minkowski space $\mathbb{M}^{n+p}$ which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition o…

The paper quantifies how much of a 4-ball must be removed to squeeze into a cylinder, proving a lower bound on the Minkowski dimension.

The study finds hypersurfaces with constant scalar curvature in Minkowski space.

Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 or one of whose kernels has dimension 3. We extend these results to the case of toroidal circle planes.

It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtai…

Study on hyperspheres in 4-spaces as special Riemannian manifolds.

Study spin-0 fields on n-dimensional Minkowski spacetimes, computing asymptotic charges.

The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such valuation is given by the sum of a valuation of degree of homogeneity 1 and 3. In dimens…

New insights from centro-affine geometry solve a key geometric conjecture.

The paper establishes correspondences between quaternionic spinors, Minkowski flags, and hyperbolic horospheres.

Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification o…

Stability of catenoid in 4D Minkowski space proven without symmetry assumptions.

We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of the vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich's result on the future hyperboloidal stability of Minkowski space…

Proves smoothness and star-shapedness of weak IMCF solutions in hyperbolic space.

We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is local…

We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz-Minkowski space $ł^3=(\r^3,dx_1^2+dx_2^2-dx_3^2),$ with fundamental piece having a finite number $(n+1)$ of singularities, is a real analytic manifold of dimension $3n+4.$ The underlyin…

Introduces supermanifolds and supersymmetry for mathematicians.

Estimates dimension of subsets from random samples, proving consistency.

Rigidity theorem shows massless hyperboloidal data embeds into Minkowski space.

New theorem disproves Angle Defect for super triangles.

Minkowski space is a physically important space-time for which the finding an adequate holographic description is an urgent problem. In this paper we develop further the proposal made in hep-th/0303006 for the description as a duality between Minkowski space-time and a Conformal Field Theory defined on the boundary of …

Paper proves rigidity of static manifolds and applies to metric extensions.

Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.

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 $n\geq 3$.