arXiv research

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

Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.

We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic surfaces and their compactifications.

Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.

The paper constructs complex hyperbolic 2-manifolds with one cusp.

3-manifolds are CR uniformized on spheres, proving a conjecture.

In this paper, we prove an equality which involves Reidemeister torsion, complex volume, and Zograf infinite product for hyperbolic 3-manifolds with cusps.

New hyperbolic 3-manifolds with multiple cusps are found that sound the same but look different.

We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that…

Paper finds infinite family of minimal triangulations for complex 3D shapes.

We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada's method in the cusped setting, and so in addition our pairs are fin…

The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We e…

It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic $3$-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete cl…

Continuous process closes cusps in complex algebraic surfaces.

The paper studies positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.

The paper constructs Poincaré-Einstein 4-manifolds with various cusps.

We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebro-geometric methods using Mumford's work on toroidal compactifications and its generalization due to N. Mok and W.-K. To, we get a bound which is considerably better than those obtained previousl…

Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.

Four hyperbolic 24-cell 4-manifolds with one cusp are identified.

The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.

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

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

4 flat 3-manifolds realized in hyperbolic 4-space.

We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume $v_m = 4π^2/3$ and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has lowest volume among…

Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing d…

In this paper we deepen the analysis of certain classes M_{g,k} of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of M_{g,k} is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusp…

This thesis investigates cusp cross-sections of arithmetic real, complex, and quaternionic hyperbolic $n$--orbifolds. We give a smooth classification of these submanifolds and analyze their induced geometry. One of the primary tools is a new subgroup separability result for general arithmetic lattices.

The paper bounds Betti numbers of complex-hyperbolic manifolds.

Study Dirac operator on cusped hyperbolic manifolds, finding spectrum properties.

We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus g. Manifolds in M_{g,k} have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In additi…

The study identifies flat manifolds with unique cusp cross-sections in arithmetic hyperbolic manifolds.

Graded identities for hyperbolic surfaces with cusps and cone points.

We prove a bound relating the volume of a curve near a cusp in a hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. There are a number of consequences: we show that for …

Study of Eisenstein series linked to hyperbolic cusps.

Census of 10-tetrahedra hyperbolic 3-manifolds with 150,730 new examples.

Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling w…

We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds wit…

This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also sho…

4-manifolds show every flat 3-manifold as cusp sections.

This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orb…

Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deform…

We introduce a generalization of the Dijkgraaf-Witten invariants for cusped or compact oriented 3-manifolds. We show that the generalized DW invariants distinguish some pairs of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same Turaev-Viro invariants. We also present an example of a pair …

Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.

The study analyzes spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.

Study shows arithmetic properties of specific hyperbolic Dehn fillings.