Related papers: Perelman's collapsing theorem for 3-manifolds

In this paper, we show that Gromov-Thurston's principle works for hyperbolic 3-manifolds of infinite volume and with finitely generated fundamental group. As an application, we have a new proof of Ending Lamination Theorem. Our proof…

Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular…

Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends…

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry groups Isom(M), determine which elliptic…

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and…

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform…

We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using…

This paper proves the Giroux Correspondence in dimension three using Heegaard splittings of contact manifolds. In two of the authors earlier paper they proved the Giroux Correspondence for tight contact 3-manifolds via convex Heegaard…

We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker…

Suppose $M$ is a closed, connected, orientable, \irr\ \3m\ such that $G=\pi_1(M)$ is infinite. One consequence of Thurston's geometrization conjecture is that the universal covering space $\widetilde{M}$ of $M$ must be \homeo\ to $\RRR$.…

We show that three natural decision problems about links and 3-manifolds are computationally hard, assuming some conjectures in complexity theory. The first problem is determining whether a link in the 3-sphere bounds a Seifert surface with…

We continue the program of Chinea, De Leon and Marrero who studied the topology of cosymplectic manifolds. We study 3-cosymplectic manifolds which are the closest odd-dimensional analogue of hyper-Kaehler structures. We show that there is…

The purpose of this paper is to give an application of the gluing theorem for special Lagrangian submanifolds of a Calabi-Yau 3-fold. We proved a gluing theorem before to smooth a codimension-two singularity of a particular special…

For each closed oriented 3-manifold $M$ in Thurston's picture, the set of degrees of self-maps on $M$ is given.

We study 3d $\mathcal{N}=2$ supersymmetric gauge theories on closed oriented Seifert manifold---circle bundles over an orbifold Riemann surface---, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our…

We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It…

We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture…

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting…

We develop the deformation theory of Calabi-Yau threefolds, by which we mean 3-dimensional complex manifolds with a nowhere-vanishing holomorphic 3-form, on manifolds with boundary. The boundary data is a closed, real 3-form on the…

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and…