Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.
problem Existence and classification of conical Calabi-Yau structures on horospherical cones.
method Variational approach to establish equivalence between volume minimization and existence of conical Calabi-Yau structures.
result Existence of many irregular horospherical cones with mild singularities.
The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.
problem Characterizing constant mean curvature surfaces in a three-dimensional light cone.
method Analyzing entire constant mean curvature graphs in the light cone Q+3 under the condition of bounded Gaussian curvature. result Entire constant mean curvature graphs in the light cone are either horospheres or spheres.
New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
Anosov groups in rank ≤3 have unique ergodic horospherical actions.
problem Characterizing unique ergodicity of horospherical actions for Anosov groups.
method Analyzing N-action on Rv for v in the limit cone. result Unique ergodicity holds for r≤3 and v in the limit cone. New non-existence results for harmonic maps into perturbed cones.
problem Proper harmonic maps into perturbed cones in \(\mathbb{R}^n\), horospheres in \(\mathbb{H}^n\).
method Extension of foliated maximum principle to non-compact settings.
result New non-existence results for proper harmonic maps.
Geometric correspondence between spinors and horospheres in hyperbolic space.
problem Understanding the relationship between spinors and horospheres in hyperbolic geometry.
method Detailed exposition and step-by-step construction of the spinor--horosphere correspondence.
result Spinor--horosphere correspondence is a smooth, SL(2,C)-equivariant bijection. Study of line bundles on spherical varieties leads to Calabi-Yau metrics.
problem Understanding linearized line bundles on spherical varieties.
method Formulas for valuative invariants and application to Fano spherical varieties.
result Calabi-Yau metrics on spherical varieties' cone.
The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
problem Existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
method Explicit K-stability condition, degeneration, and asymptotic cone analysis.
result Uniqueness of K-invariant Calabi-Yau metrics on affine spherical manifolds. A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighb…
Extends spinor-horosphere correspondence to higher dimensions and new spinor types.
problem Constructing isomorphisms between spinors and horospheres in hyperbolic spaces.
method Generalizes Mathews' results to Lipschitz spinors and null multiflags in higher-dimensional hyperbolic spaces.
result Equivariant correspondence between two-component Lipschitz spinors and higher-dimensional horospheres.
Study on volume growth of horospheres in specific Heintze groups.
problem Volume growth of horospheres in diagonalizable Heintze groups.
method Analysis of volume growth in Heintze groups with diagonalizable structure.
result Two distinct isometry and quasi-isometry classes of horospheres with explicit volume growth calculations.
The paper examines stable capillary hypersurfaces in hyperbolic space.
problem Stability of capillary hypersurfaces with free boundary on a horosphere.
method Analysis of umbilical and totally geodesic hypersurfaces using stability criteria.
result Umbilical and totally geodesic hypersurfaces are the only stable capillary hypersurfaces with boundary on a horosphere.
New formulas for hyperbolic mass using horospheres.
problem Mass calculation of asymptotically hyperbolic manifolds.
method Geometric formulas derived using coordinate horospheres.
result Improved rigidity results of hyperbolic space.
The study proves properties of intersections of horospheres in harmonic spaces.
problem Properties of intersections of horospheres in harmonic spaces.
method Constructing volume preserving mappings using Busemann functions.
result Upper bound of the volume of intersection of horospheres is independent of Busemann function differences.
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
Study measures invariant under horospherical subgroups for finitely generated Kleinian groups.
problem Investigating measures invariant under horospherical subgroups for finitely generated Kleinian groups.
method Combining results from Landesberg and Lindenstrauss with 3-manifold theory, including the Tameness Theorem.
result Identified all Radon measures on quotient space that are ergodic and invariant under horospherical subgroup.
Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.
problem No universally accepted sum for sets in hyperbolic space.
method Introduces hyperbolic p-sum and develops horospherical p-Brunn-Minkowski theory.
result Solves p-Minkowski and p-Christoffel-Minkowski problems for various p.
We study the geometry of horospheres in Teichmüller space of Riemann surfaces of genus g with n punctures, where 3g−3+n≥2. We show that every C1-diffeomorphism of Teichmüller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and…
Corrected proof for 3D harmonic manifolds with minimal horospheres.
problem Proving 3D harmonic manifolds with minimal horospheres are either flat or hyperbolic.
method Provided a corrected proof for the classification of 3D harmonic manifolds.
result Classification of 3D harmonic manifolds: flat or hyperbolic.
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…
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing th…
Study on hyperbolic manifolds with special boundaries.
problem Characterizing geometric properties of hyperbolic manifolds with specific boundaries.
method Analyzing asymptotically hyperbolic manifolds with a horospherical boundary.
result Derived geometric formulas for the model case of horoballs and their complements.
We use Bryant Representation to construct constant mean curvature one surfaces in hyperbolic space that desingularize a horosphere packing.
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.
problem Understanding geometric relationships between spinors and horospheres in hyperbolic space.
method Explicit bijective correspondence between spinors and horospheres, using bilinear forms and complex-valued distances.
result Derived applications include Ptolemy equations and Plücker coordinates connections.
Study on horospheres in higher rank homogeneous spaces, proving density properties.
problem Density of horospheres in higher rank homogeneous spaces.
method Analyzing maximal horospherical subgroups and their minimal subsets in the context of Furstenberg boundary.
result Equivalence of horospherical limit points and density properties in higher rank homogeneous spaces.
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.
problem Understanding the rigidity of quasi-isometries in specific geometric spaces.
method Proving quasi-isometries are uniformly close to product maps in horospherical products of hyperbolic spaces.
result Quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid.
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.
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 …
Study calculates Ricci bounds for special Fano manifolds.
problem Computing Ricci bounds for specific Fano manifolds.
method Using barycenter of moment polytopes with Duistermaat-Heckman measure.
result Greatest Ricci lower bounds can be arbitrarily close to zero.
Paper proposes a new classifier for hyperbolic spaces using horospherical boundaries.
problem Optimization of large margin classifiers in hyperbolic spaces.
method Horospherical decision boundaries for geodesically convex optimization.
result Geodesically convex optimization leads to globally optimal solutions.
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…
The paper proves rigidity and ergodicity of horospherical foliations.
problem Rigidity and ergodicity of horospherical foliations in higher rank.
method Establishes higher rank extensions of rigidity theorems for representations of discrete subgroups of divergence type, using conformal measures and boundary maps.
result Proves conformal measure rigidity and ergodicity of horospherical foliations for hypertransverse subgroups.
A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on S2 with curvature K>−1 is induced on a unique convex surface in H3. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another …
Paper solves Christoffel-Minkowski problem in hyperbolic space.
problem Prescribing k-th horospherical p-surface area measure of h-convex domains in hyperbolic space. method Considered a fully nonlinear equation and used the full rank theorem with a viscosity approach.
result Existence of uniformly h-convex solution under appropriate assumptions. 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.
This paper studies convergence of horospheres in CAT(0) spaces.
problem Analysis of convergence of horospheres in CAT(0) spaces.
method Examines horofunctions associated with sublinearly contracting geodesic rays.
result Horospheres associated with sublinearly contracting horofunctions are convergent.
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 …
The paper establishes correspondences between quaternionic spinors, Minkowski flags, and hyperbolic horospheres.
problem Understanding geometric correspondences in 4D hyperbolic geometry.
method Explicit bijective correspondences using Clifford matrices and bilinear forms.
result Lambda lengths generalize to quaternionic values in 4D hyperbolic space and satisfy a non-commutative Ptolemy equation.
The study connects lamination and orbit closures in hyperbolic manifolds.
problem Understanding the geometric and dynamical properties of horocycle orbit closures in Z-covers of compact hyperbolic manifolds. method Exposes connections between distance minimizing laminations and horospherical orbit closures in Z-covers of compact hyperbolic manifolds. Provides novel constructions and explicit descriptions. result Even slight perturbations to hyperbolic metrics can drastically change horocycle orbit closures.
This paper classifies solutions to a specific hyperbolic geometry problem.
problem Classifying solutions to a specific hyperbolic geometry equation.
method Analytical and numerical methods to solve the equation.
result Classification of solutions for p≥−7, nonuniqueness for p<−7. Paper solves Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
problem Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
method Proved existence of solutions using a new full rank theorem.
result Existence of smooth, origin-symmetric, strictly horospherically convex solutions.