arXiv research

The paper examines stable capillary hypersurfaces in hyperbolic space.

Study on hyperbolic manifolds with special boundaries.

Paper proposes a new classifier for hyperbolic spaces using horospherical boundaries.

In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationa…

This paper studies convergence of horospheres in CAT(0) spaces.

Study on horospheres in higher rank homogeneous spaces, proving density properties.

Study classifies and characterizes translators in hyperbolic static universe.

Paper finds eigenvalue bounds for hyperbolic space domains.

In the early 80's S.-T. Yau posed the problem of establishing the rigidity of the Hawking-Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets - the horospheres. The regularity theory in the Lorentzian …

A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex surface in $H^3$. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another …

The paper proves rigidity and ergodicity of horospherical foliations.

Develops Aleksandrov reflection for hyperbolic flows, proving convergence to umbilic surfaces.

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …

We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the `achronal limits' introduced in [12]. This, in turn, allows for a broa…

A new depth measure and median defined on Hadamard manifolds.

Extends spinor-horosphere correspondence to higher dimensions and new spinor types.

Study on volume growth of horospheres in specific Heintze groups.

New formulas for hyperbolic mass using horospheres.

The study proves properties of intersections of horospheres in harmonic spaces.

Geometric correspondence between spinors and horospheres in hyperbolic space.

Conditions ensure constant curvature in negatively curved manifolds.

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

Study measures invariant under horospherical subgroups for finitely generated Kleinian groups.

Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.

We study the geometry of horospheres in Teichmüller space of Riemann surfaces of genus g with n punctures, where $3g-3+n\geq 2$. We show that every $C^1$-diffeomorphism of Teichmüller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and…

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

Corrected proof for 3D harmonic manifolds with minimal horospheres.

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…

We use Bryant Representation to construct constant mean curvature one surfaces in hyperbolic space that desingularize a horosphere packing.

The paper characterizes subgroup stability via limit sets on the Morse boundary.

Let (M,g) be a non-compact and complete Riemannian manifold with minimal horospheres and infinite injectivity radius. We prove that bounded functions on (M,g) satisfying the mean-value property are constant. We extend thus a result of A. Ranjan and H. Shah who proved a similar result for bounded harmonic functions on h…

We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon inversion formula. In a more broad context, this possibility reflects the fact that the …

This paper connects spinors to horospheres in hyperbolic space.

Paper solves a mixed boundary value problem in space forms with umbilical boundaries.

We study horospheres in hyperbolic 3-manifolds $M$ all whose ends are degenerate. Towards this, we study which almost minimizing geodesics in $M$ go through arbitrarily thin parts.

Quasi-isometries in horospherical products are close to product maps.

We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclus…

In this paper, we extend the result about the existence of Kähler-Ricci soliton on toric manifold (proved by Wang and Zhy) by proving this existence on horospherical varieties using the continuity method.

Suppose X/Gamma is an arithmetic locally symmetric space of noncompact type (with the natural metric induced by the Killing form of the isometry group of X), and let p be a point on the visual boundary of X. It was shown by T.Hattori that if each horoball based at p intersects every Gamma-orbit in X, then p is not on t…

Study calculates Ricci bounds for special Fano manifolds.

This paper studies surfaces in hyperbolic space using horospheres and proves existence and regularity theorems.

Paper solves Christoffel-Minkowski problem in hyperbolic space.

In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.

Anosov groups in rank ≤3 have unique ergodic horospherical actions.

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both …

We prove that the existence of one flat horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension n with n greater than or equal to 3, implies that the manifold is homothetic to a real hyperbolic manifold.

In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.

In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded …