arXiv research

New interpretation of complex hyperbolic form as Weil-Petersson form.

Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.

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

Study examines Kähler immersions of ALE Kähler metrics into complex space forms.

We show that an isoparametric submanifold of a complex hyperbolic plane, according to the definition of Heintze, Liu and Olmos', is an open part of a principal orbit of a polar action. We also show that there exists a non-isoparametric submanifold of the complex hyperbolic plane that is isoparametric according to the d…

We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.

Classifies actions on complex space forms with Lagrangian orbits.

Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.

This paper connects spinors to horospheres in hyperbolic space.

Study warped product metrics on hyperbolic and complex hyperbolic manifolds.

We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth…

The paper explores properties of projections and gradient methods in hyperbolic space forms.

The paper studies Lagrangian surfaces in a specific Riemannian product space.

Study on complex hyperbolic bidisk isometries and their Dirichlet domains.

This paper studies deformations of hyperbolic surfaces with special structures.

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.

We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic groups and end with some open questions.

The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and co…

This paper constructs Poisson transforms and analyzes their properties on complex hyperbolic spaces.

Generalizes Leighton's theorem to cube complexes.

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural fo…

We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact $1$-forms $λ_1^*$ on rational homology spheres…

The paper bounds Betti numbers of complex-hyperbolic manifolds.

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…

We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous sp…

The paper classifies discrete complex hyperbolic triangle groups.

Sharp asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.

This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in case…

Geometric structures over algebras describe geodesics and spaces.

Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of…

In this paper we study the homogeneous Kaehler manifolds (h.K.m.) which can be Kaehler immersed into finite or infinite dimensional complex space forms. On one hand we completely classify the h.K.m. which can be Kaehler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. Moreover, we e…

It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic spac…

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary num…

We show that ruled real hypersurfaces with constant mean curvature in the complex projective and hyperbolic spaces must be minimal. This provides their classification, by virtue of a result of Lohnherr and Reckziegel.

New examples of real hypersurfaces found in complex hyperbolic quadrics.

Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to …

Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(φ, ξ, η, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_ξ=R(\cdot,ξ)ξ$ is $ξ$-parallel. In particular, we prove that the condition $\nabla_ξ R_ξ=0$ characterizes the …

The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.

It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb {CH}^2$…

Study topological hyperbolicity of moduli spaces of elliptic surfaces.

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…

Study Lagrangian submanifolds on complex hyperbolic quadric.

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomegeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH^n foliated by umbilical hypersurfaces of Lagrangian s…

We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabo…

This paper extends, to a class of systems of semi-linear hyperbolic second order PDEs in three variables, the geometric study of a single nonlinear hyperbolic PDE in the plane as presented in [Anderson I.M., Kamran N., Duke Math. J. 87 (1997), 265-319]. The constrained variational bi-complex is introduced and used to d…

Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $Γ$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of …