Thurston Manifold Research Articles

Left-invariant almost complex structures on the higher dimensional Kodaira–Thurston manifolds

We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira–Thurston manifolds.

Bott–Chern and bar{partial } harmonic forms on almost Hermitian 4-manifolds

We prove that on a compact almost Hermitian 4-manifold the space of {bar{partial }}-harmonic (1, 1)-forms always has dimension h_{{bar{partial }}}^{1,1} = b_- +1 or b_-, whilst the space of Bott–Chern harmonic (1, 1)-forms always has dimension h_{BC}^{1,1} = b_- +1. We also perform calculations of h^{2,1}_{BC} and h^{1,2}_{BC} on the Kodaira–Thurston manifold, thereby providing a full account of when h^{p,q}_{BC} is or is not invariant of the choice of almost Hermitian metrics. Finally, we introduce a decomposition of the space of L^2 functions on all torus bundles over S^1, which has proven useful for solving linear PDEs, and we demonstrate its use in the calculation of h^{p,q}_{{bar{partial }}}.

Bott–Chern Laplacian on almost Hermitian manifolds

Let (M,J,g,omega ) be a 2n-dimensional almost Hermitian manifold. We extend the definition of the Bott–Chern Laplacian on (M,J,g,omega ), proving that it is still elliptic. On a compact Kähler manifold, the kernels of the Dolbeault Laplacian and of the Bott–Chern Laplacian coincide. We show that such a property does not hold when (M,J,g,omega ) is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira–Thurston manifold. Furthermore, if (M,J,g,omega ) is a connected compact almost Hermitian 4-manifold, denoting by h^{1,1}_{BC} the dimension of the space of Bott–Chern harmonic (1, 1)-forms, we prove that either h^{1,1}_{BC}=b^- or h^{1,1}_{BC}=b^-+1. In particular, if g is almost Kähler, then h^{1,1}_{BC}=b^-+1, extending the result by Holt and Zhang (Harmonic forms on the Kodaira–Thurston manifold. arXiv:2001.10962, 2020) for the kernel of Dolbeault Laplacian. We also show that the dimensions of the spaces of Bott–Chern and Dolbeault harmonic (1, 1)-forms behave differently on almost complex 4-manifolds endowed with strictly locally conformally almost Kähler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the Bott–Chern cohomology groups for almost complex manifolds, recently introduced in Coelho et al. (Maximally non-integrable almost complex structures: an h-principle and cohomological properties, arXiv:2105.12113, 2021).

On the Anti-invariant Cohomology of Almost Complex Manifolds

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non-compact. We construct families of (non-integrable) almost complex structures on $${{\mathbb {R}}}^4$$ , such that the space of closed J-anti-invariant forms is infinite dimensional, and also 0- or 1-dimensional. In the compact case, we construct 6-dimensional almost complex manifolds with arbitrary large anti-invariant cohomology and a 2-parameter family of almost complex structures on the Kodaira–Thurston manifold whose anti-invariant cohomology group has maximum dimension.

Cubulation of Gromov–Thurston manifolds

In this article we prove that the fundamental group of certain manifolds, introduced by Gromov and Thurston [GT87] and obtained by branched cyclic covering over arithmetic manifolds, acts geometrically on a CAT (0) cube complex. We show in particular that these groups are linear over \mathbb Z .

Partial integrability of almost complex structures on Thurston manifolds

We prove that any left-invariant symplectic almost complex structure on a Thurston manifold which is compatible with its canonical left-invariant Riemannian metric has holomorphic type 1.

The Calabi–Yau equation on the Kodaira–Thurston manifold, viewed as an $S^1$-bundle over a 3-torus

We prove that the Calabi–Yau equation on the Kodaira–Thurston manifold has a unique solution for every $S^1$-invariant initial datum.

Pseudoholomorphic tori in the Kodaira–Thurston manifold

The Kodaira–Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov–Witten invariants which count pseudoholomorphic tori in the Kodaira–Thurston manifold. For a fixed symplectic form the Gromov–Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov–Witten computation for a non-Kähler manifold.

Two-sided bounds for the complexity of hyperbolic three-manifolds with geodesic boundary

We construct an infinite family of hyperbolic three-manifolds with geodesic boundary that generalize the Thurston and Paoluzzi-Zimmermann manifolds. For the manifolds of this family, we present two-sided bounds for their complexity.

Teichmüller space of negatively curved metrics on Gromov–Thurston manifolds is not contractible

In this paper we prove that for all n = 4k - 2, k ≥ 2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that [Formula: see text] is nontrivial. [Formula: see text] denotes the Teichmüller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity. Gromov–Thurston branched cover manifolds provide examples of negatively curved manifolds that do not have the homotopy type of a locally symmetric space. These manifolds will be used in this paper to prove the above stated result.

Solvable Models for Kodaira Surfaces

In this work, we study families of compact spaces which are of the form \({G/\Lambda_{k,i}}\) for G the oscillator group and \({\Lambda_{k,i} < G}\) a lattice. The solvmanifolds \({G/\Lambda_{k,i}}\) are not pairwise diffeomorphic and one has the coverings \({G \to M_{k, 0} \to M_{k, \pi} \to M_{k, \pi/2}}\) for \({k \in \mathbb{Z}}\) . We compute their cohomologies and minimal models. Each manifold Mk, 0 is diffeomorphic to a Kodaira–Thurston manifold, i.e., a compact quotient \({S^1 \times {\rm H}_3 (\mathbb{R}) /\Gamma_k}\) where \({\Gamma_k}\) is a lattice of the real three-dimensional Heisenberg group \({{\rm H}_3 (\mathbb{R})}\) . Furthermore, any Mk, 0 provides an example of a solvmanifold whose cohomology does not depend on the Lie algebra only. We explain some geometrical aspects of those compact spaces, to show how to distinguish them (by invariant complex, symplectic and metric structures). For instance, no invariant symplectic structure of G can be induced to the any quotient.

Partial integrability on Thurston manifolds

We determine the maximal number of independent holomorphic functions on the Thurston manifolds $M^{2r+2}$, $r\geq 1$, which are the first discovered compact non-Kähler almost Kähler manifolds. We follow the method which involves analyzing the torsion tens

EXAMPLES OF COMPACT NEGATIVELY CURVED ALMOST COMPLEX, BUT NOT COMPLEX, MANIFOLDS

In this paper, we provide infinitely many examples of compact negatively curved almost complex, but not complex, manifolds of complex dimension 2n + 1 or 2n + 2 by using strongly pseudoconvex homogeneous domains in an almost complex manifold. Unlike the Kodaira–Thurston manifold which is the flat case, the first Betti number of the constructed manifolds is even, and their first homology group is, in fact, isomorphic to ℤ4n+2 or ℤ4n+4.

The Calabi–Yau equation on the Kodaira–Thurston manifold

AbstractWe prove that the Calabi–Yau equation can be solved on the Kodaira–Thurston manifold for all given T2-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic 4-manifolds with compatible but non-integrable almost complex structures.

Theta functions on the Kodaira–Thurston manifold

The Kodaira—Thurston manifold M is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kahler. We describe a construction of ϑ-functions associated to M, which parallels the classical theory of ϑ-function associated to the torus (from the point of view of representation theory and geometry), and which yields pseudoperiodic complex-valued functions on ℝ 4 . There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira—Thurston manifold and on the universal cover of M in a Hamiltonian fashion. The ϑ-functions discussed in this paper are intimately related to the representation theory of G in much the same way that the classical ϑ-functions are related to the Heisenberg group. One aspect of our results is a connection between the representation theory of G and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M; in particular, we show that G-invariant special Lagrangian foliations can be detected by a simple algebraic condition on certain subalgebras of the Lie algebra of G. Crucial to our generalization of ϑ-functions is the spectrum of the Laplacian Δ acting on sections of certain line bundles over M. One corollary of our work is a verification of a theorem of Guillemin-Uribe describing the structure (in the semiclassical limit) of the low-lying spectrum of Δ.

Theta functions on the Kodaira-Thurston manifold

We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).

Convex projective structures on Gromov–Thurston manifolds

We consider Gromov-Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n some of the Gromov-Thurston manifolds admit strictly convex real-projective structures.

Computation of eigenmodes on a compact hyperbolic 3-space

Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace - Beltrami operator. Using the direct boundary element method, we obtain numerically the low-lying eigenmodes on a compact hyperbolic 3-space called the Thurston manifold which is the second smallest in the known compact hyperbolic 3-manifolds. The computed eigenmodes are expanded in terms of eigenmodes on the unit three-dimensional pseudosphere. We find numerically that the expansion coefficients behave as Gaussian pseudo-random numbers for low-lying eigenmodes. The observed Gaussianity in the CMB fluctuations can partially be attributed to the Gaussian pseudo-randomness of the expansion coefficients assuming that the Gaussian pseudo-randomness is the universal property of the compact hyperbolic spaces.