Related papers: A model for random three--manifolds

We prove an extension of Milnor-Wood inequalities to a geometric situation. We study representations of the fundamental group of a compact manifold into the isometry group of a product of rank one spaces of the same dimension and show an…

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…

In this paper, we study the finite boundary regularity and estimates of an asymptotically hyperbolic Einstein manifold in even dimension $n+1.$ We show that if the initial compactification is $C^{n-1}$ and the $(n-3)$-th derivative of its…

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and…

It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by…

We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical…

We show that a smooth embedding of a closed 3-manifold in S^3 x R can be isotoped so that every generic level divides S^3 x t into two handlebodies (i.e., is Heegaard) provided the original embedding has a unique local maximum with respect…

Spectrahedra are affine-linear sections of the cone $\mathcal{P}_n$ of positive semidefinite symmetric $n\times n$-matrices. We consider random spectrahedra that are obtained by intersecting~$\mathcal{P}_n$ with the affine-linear space…

We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifolds and the Fibonacci manifolds.

The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric.…

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds…

An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double…

It is natural to ask how many isotopy classes of embedded essential surfaces lie in a given 3-manifold. The first bounds on the number of such surfaces were exponential, using normal surfaces. More recently, by restricting to alternating…

This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov--Pyatetski-Shapiro and…

We prove a smooth analogue of the classical Thom-Milnor bound, showing that the Betti numbers of the zero set of a smooth map on a compact Riemannian manifold can be controlled by a condition number computed from its first jet. This extends…

We show that for every sequence $(n_i)$, where each $n_i$ is either an integer greater than 1 or is $\infty$, there exists a simply connected open 3-manifold $M$ with a countable dense set of ends $\{e_i\}$ so that, for every $i$, the genus…

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses…

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample…

In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary minimal two-disk, which locally maximizes a modified Hawking mass, is embedded in a…

Kevin Hartshorn showed that if a three-dimensional manifold $M$ admits a Heegaard surface $\Sigma$ with Hempel distance $d$ then every incompressible surface in $M$ has genus at least $\frac{d}{2}$. Scharlemann-Tomova generalized this,…