arXiv research

Compact hyperbolic complex manifolds are rigid under deformation.

Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.

Study properties of balanced hyperbolic compact complex manifolds.

The paper defines two types of hyperbolicity for complex manifolds and proves related results.

The paper builds complex hyperbolic 2-manifolds with isolated singularities.

Compactify complex hyperbolic almost Hermitian manifolds.

Proves hyperbolized groups are virtually compact special and linear.

Compactifies CR structures for complex hyperbolic manifolds.

Study shows infinite manifold types for every group.

The paper bounds Betti numbers of complex-hyperbolic manifolds.

Extends Fried's result to arbitrary representations of compact hyperbolic manifolds.

Introduces new hyperbolicity concepts for complex manifolds.

Study bounds changes in hyperbolic 3-manifold structures after drilling short geodesics.

Scharlemann and Thompson define a numerical complexity for a 3-manifold using handle decompositions of the manifold. We show that for compact hyperbolic 3-manifolds this is linearly related to a definition of metric complexity in terms of the areas of level sets of Morse functions.

Study of flows on 7D manifolds with holomorphic properties.

Study of flows on complex manifolds with holomorphic properties.

Study of hyperbolic behavior in complex manifolds with specific vector bundles.

We describe a natural strategy to enumerate compact hyperbolic 3-manifolds with geodesic boundary in increasing order of complexity. We show that the same strategy can be employed to analyze simultaneously compact manifolds and finite-volume manifolds having toric cusps. In opposition to this we show that, if one allow…

Researchers prove the arc complexes of decorated hyperbolic polygons are balls.

Compact manifolds with specific cover properties are hyperbolic.

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of th…

Study deformations of compact Kähler hyperbolic manifolds.

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on o…

We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodes…

Using the Donaldson-Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfa…

A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is…

Consider a compact Kähler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $$ has maximal bottom of spectrum $λ_1(\widetilde{M}$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.$

In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles…

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volu…

Survey on recent breakthrough linking curvature and Kobayashi hyperbolicity.

We prove the Bloch conjecture : $ c_2(E) \in H^4_\cald (X,\bbz(2))$ is torsion for holomorphic rank two vector bundles $E$ with an integrable connection over a complex projective variety $X$. We prove also the rationality of the Chern-Simons invariant of compact arithmetic hyperbolic three-manifolds. We give a sharp hi…

The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$, thereby establishing that these classes of manifolds have the same growth rate w…

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric …

Study on new hyperbolicity notions for non-Kähler manifolds and their deformations.

New examples of real hypersurfaces found in complex hyperbolic quadrics.

Unique hyperbolic manifolds identified by boundary pleating.

No spin structures found in a hyperbolic 4D space.

Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic Kähler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound i…

Study Brownian motion on Grassmann manifold using matrix stochastic calculus.

Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in w…

Conditions for trivial gradient hyperbolic Ricci and Yamabe solitons to be Einstein or constant scalar curvature.

Study on Cheeger constant in specific hyperbolic 3-manifolds.

Paper proves stability of positive mass theorem for specific types of manifolds.

New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.

The study shows how certain surfaces can be filled by hyperbolic manifolds.

We show in this article that Kähler hyperbolic manifolds satisfy a family of optimal Chern number inequalities and the equality cases can be attained by some compact ball quotients. These present restrictions to complex structures on negatively-curved compact Kähler manifolds, thus providing evidence to the rigidity co…

We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most πcan be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local rigidity by Hodgson and Kerckhoff implies the global rigidity for compact orientable hype…

The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.