arXiv research

Conditions ensure constant curvature in negatively curved manifolds.

New formulas for hyperbolic mass using horospheres.

Corrected proof for 3D harmonic manifolds with minimal horospheres.

Study on hyperbolic manifolds with special boundaries.

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…

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…

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

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

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.

Study calculates Ricci bounds for special Fano manifolds.

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 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.

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.

The study examines continuous mean curvature functions on manifolds without conjugate points.

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.

The study connects lamination and orbit closures in hyperbolic manifolds.

New method for optimization on Hadamard manifolds with curvature-independent guarantees.

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 …

New isoparametric hypersurfaces found in Damek-Ricci spaces.

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 …

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

A ${ m G}_2$-horospherical manifold is identified by its VMRT.

Study on volume growth of horospheres in specific Heintze groups.

The paper examines stable capillary hypersurfaces in hyperbolic space.

Researchers study horospherical transforms on hyperboloids.

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…

New Calabi-Yau metrics found on complex symmetric spaces.

A new depth measure and median defined on Hadamard manifolds.

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds $X$ with mild curvature boundedness c…

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fun…

Researchers characterize a specific type of projective variety based on its tangents.

Abelian covers of hyperbolic $3$-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic $3$-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as…

Geometric correspondence between spinors and horospheres in hyperbolic space.

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

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.

Let $(M,g)$ be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that $M$ is a flat manifold, provided $M$ is asymptotically harmonic of constant $h = 0$.

Horospheres, hyperspheres, and hyperplanes in hyperbolic spaces are rigid in terms of mean curvature.

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

This paper connects spinors to horospheres in hyperbolic space.

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

We investigate submanifolds in space forms such that every geodesic orthogonal to the submanifold intersects a fixed totally geodesic submanifold. We obtain an application to horospheres in Hadamard manifolds.

New method optimizes on curved manifolds without curvature dependence.

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

New hyperbolic sliced-Wasserstein distances derived for efficient comparison.

$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\{…

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…