arXiv research

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.

Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.

New conditions found for hyperbolic bicycle tracks.

One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.

Study finds optimal loops in hyperbolic space with Finsler structure.

The paper examines flows that preserve area and length in hyperbolic geometry.

Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$. In this paper we show that any geodesic plane in $M^*$ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theore…

The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.

Let $M$ be a geometrically finite acylindrical hyperbolic 3-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained earlier by McMullen, Moh…

New translations defined; curve shortening flow solved in hyperbolic plane.

A discrete method approximates hyperbolic curvature flow in the plane.

Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.

Hot spots conjecture proven for small eigenvalue domains.

Geometric approach solves maximum likelihood for Cauchy-like distributions.

No accelerated gradient method for hyperbolic convex functions.

Proof of Graustein's theorem in different geometries.

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

We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero

The paper studies a flow of spacelike surfaces in Lorentz-Minkowski space, proving convergence to a hyperbolic plane.

Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.

The renormalized volume is reinterpreted using isoperimetric profiles.

The paper studies how certain spacelike surfaces evolve over time in a specific space.

We describe in this paper a geometric construction in the projective p-adic plane that gives, together with a suitable notion of p-adic convexity, some open subsets of P 2 .Q p / naturally endowed with a "Hilbert" distance and a transitive action of PGL.2; Q p / by isometries. ese open sets are natural analogues of the…

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with pres…

We consider a convex curve $γ$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $γ$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curv…

We prove that any weakly acausal curve $Γ$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$. The curve $Γ$ is the graph of a quasisymmetric homeomorphism of the circle if and only…

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function al…

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed…

Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that $C^2$ closed convex curves possess at least two tripod configurations. Later, Kao and Wang established the existence of tripod configurations for $C^2$ closed locally c…

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Co…

New patterns on spheres and hyperbolic planes described by integrable systems.

We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in…

Constructs harmonic maps near retractions in hyperbolic spaces.

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.

The paper explores centroids and static equilibrium points in non-Euclidean geometries.

On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…

The paper solves optimal control problems for various convex sets using convex trigonometry.

The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.

New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.

We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant $b$ on the space of those isometric deformations which, for conv…

Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbol…

In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…

An exotic plane exists in an acylindrical 3-manifold without being closed.

The paper studies how spacelike surfaces evolve in Lorentz-Minkowski space over time.