arXiv research

Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.

Study cohomology of ball quotients and their compactifications.

We first show that for a bounded pseudoconvex domain with a manifold quotient of finite-volume in the sense of Kahler-Einstein measure, the identity component of the automorphism group of this domain is semi-simple without compact factors. This partially answers an open question in [Fra95]. Then we apply this result in…

A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial compu…

A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then …

Linear bound on Betti numbers of negatively curved orbifolds.

Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …

Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar cur…

We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riem…

Geodesics in curved spaces spread evenly over time.

Study on curvature properties and Shafarevich conjecture for complex hyperbolic manifolds.

In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our…

The study classifies Riemannian manifolds with curvature nullity.

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of c…

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $π_1(M)$.…

We characterize the universal covering of connected analytic pseudo-Riemannian manifolds which admit a non-trivial and isometric action of the simple Lie group $SL(3,\mathbb{R})$ with a dense orbit preserving a finite volume. If such manifold is also weakly irreducible we prove that $M$ is isometric to, or a quotient s…

Building on results of Arthur and Mok, we extend to (finite volume) complex and quaternionic hyperbolic manifolds the results of arXiv:1004.1085. For the spherical spectrum our results are optimal. Finally, as an application we prove a Lefschetz property for the restriction map between arithmetic quotients of complex b…

We obtain restrictions on the topology of a closed connected manifold B that bounds a (possibly noncompact) manifold whose interior V admits a complete Riemannian metric of nonpositive sectional curvature. If G denotes the fundamental group of B, then a sample result is that B must be aspherical and incompressible if o…

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, ω_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positi…

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton's compactness theorem that does not assume a l…

We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $Γ$. If the quotient manifold $\bar X= Γ\backslash X$ is asymptotically $1/4$-pinched, we prove that $Γ$ is divergent and $U\bar X$ has finite Bowen-Margulis measure (which is t…

We give a classification of many closed Riemannian manifolds M whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds $M$ such that Isom$(\widetilde{M})$ has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bund…

Orbifold uniformization of complex algebraic variety via polystable parabolic Higgs bundle

Finite volume ends found in quaternionic Kähler manifolds.

Let $M = \mathbb{B}^2 / Γ$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ h…

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric sp…

Affine deformations of convex cones on projective surfaces.

Let M be a complete simply connected Riemannian manifold, with sectional curvature K bounded above by -1. Under some assumptions on the geometry of the boundary of M, which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M, and any point x_0 outsi…

Entropy rigidity proven for 3D and higher convex projective manifolds.

The study classifies 331 specific 4D polytopes with 7 facets.

Let $G$ be a non-compact simple Lie group with Lie algebra $\mathfrak{g}$. Denote with $m(\mathfrak{g})$ the dimension of the smallest non-trivial $\mathfrak{g}$-module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold $M$ it is observed that …

Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume…

We compute the space of $L^2$ harmonic forms (outside the middle degrees) on negatively curved Kaehler manifolds of finite volume.

Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime…

Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region $U$ can be swept out by a family of hypersurfaces of volume at most $V$, then it can be swept out by a fa…

The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.

A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein…

We show that complete uniform visibility manifolds of finite volume with sectional curvature $-1 \leq K \leq 0$ have positive simplicial volumes. This implies that their minimal volumes are non-zero.

In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensor- products of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.

No spin structures found in a hyperbolic 4D space.

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $Σ$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We th…

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $ν$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two tr…

We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S^3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate o…