arXiv research

New flat Minkowski planes created from convex functions.

Study of a flow related to the Orlicz-Minkowski problem for convex hypersurfaces.

Study on convex surfaces in Minkowski space, proving completeness and incompleteness conditions.

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in Finsler manifolds and give the necessary and sufficient conditions for a transnormal function to be isoparametric. We then prove that hyperplanes, Minkowski hyperspheres and $F^*$-Minkowski cylinders in a Minkowski space with $BH$-vo…

In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére t…

Paper solves a generalized chord Minkowski problem using Gauss curvature flows.

Solves capillary $L_p$-Christoffel-Minkowski problem in half-space.

The Funk--Minkowski transform ${\mathcal F}$ associates a function $f$ on the sphere ${\mathbb S}^2$ with its mean values (integrals) along all great circles of the sphere. Thepresented analytical inversion formula reconstruct the unknown function $f$ completely if two Funk--Minkowski transforms, ${\mathcal F}f$ and ${…

The paper solves a geometric problem using curvature flow and variational methods.

The hyperbolic space $ \H^d$ can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $Γ$ is a group of linear isometries of the Minkowski space such that $\H^d/Γ$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, g…

The paper classifies isoparametric hypersurfaces in conic Finsler spaces.

New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.

Solves Minkowski problem for affine invariant convex domains.

In this paper the dual Orlicz-Minkowski problem, a generalization of the $L_p$ dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.

Paper studies inverse curvature flows and solves related geometric problems.

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.

Extends Newton's minimal resistance problem to Lorentz-Minkowski space.

The paper proves fundamental theorems for timelike surfaces in Minkowski 4-space.

Smooth even solutions found for a generalized convex geometry problem.

We prove that the set of smooth, $π$-periodic, positive functions on the unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in the set of all smooth, $π$-periodic, positive functions on the unit circle with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary condition on the solv…

Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curva…

Sharp Minkowski inequality for convex surfaces in curved spaces.

The paper introduces canonical parameters for marginally trapped surfaces in Minkowski space.

For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the classical Bernstein theorem for minimal surfaces in 3-dimensional Euclidean spa…

Proves inequality for special 3D shapes, generalizing to non-symmetric ones.

The study generalizes Minkowski inequalities for curves on surfaces.

The classical Minkowski problem in Minkowski space asks, for a positive function $φ$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $φ(η)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $η$. Analogously to the Euclidean case, it is possible to f…

A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquel…

In this paper, we introduce the pseudo-torsion functions along spacelike curves whose curvature vector field has isolated lightlike points in Lorentz-Minkowski 3-space, and prove the fundamental theorem. Moreover, we analyze the behavior of the torsion function at such points. As a corollary, we obtain a necessary and …

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the …

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our…

Solves capillary curvature problems for specific p values.

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

Unified flow solves $L^p$ Christoffel-Minkowski problem for $p>1$.

We obtain isometric minimal helicoidal and rotational surfaces using generalized Bour's theorem in three dimensional Minkowski space. In addition, we show that the surfaces preserve minimality when their Gauss maps identically equal, choosing any diffentiable functions on the profile curve.

Paper solves Christoffel-Minkowski problem in hyperbolic space.

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

New capillary Christoffel-Minkowski problem solved for half-space.

We give a geometrically intrinsic construction of a global time function for relatively compact diamond-shaped regions in arbitrary spacetimes. In the case of Minkowski spacetime, the flow of diffeomorphisms associated to a suitably normalized gradient of this time function becomes the conformal isotropy subgroup of th…

Classifies zero mean curvature surfaces in Lorentz-Minkowski space.

We prove that every $(3+1)$-dimensional flat GHMC Minkowski spacetime which is not a translation spacetime or a Misner spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In otherwords, we prove that every such spacetime carries a unique time function with isochrones of constan…

We introduce the notion of $k$-type slant helix in Minkowski space $\e_1^4$. For partially null and pseudo null curves in $\e_1^4$, we express some characterizations in terms of their curvature and torsion functions.

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…

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Ya…

The paper studies a flow of spacelike curves in a Lorentz-Minkowski plane, showing convergence to a constant function.

New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.

Given a positive function F on Sn which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in Rn+1. We give some new characteriz…