Related papers: The disjoint curve property

We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,\varphi)$, where $C$ is a compact connected Riemann surface and $\varphi : C\to X$ a holomorphic map,…

The long standing classification problem in the theory of Heegaard splittings of 3-manifolds is to exhibit for each closed 3-manifold a complete list, without duplication, of all its irreducible Heegaard surfaces, up to isotopy. We solve…

We prove that for any distance at least 3 Heegaard splitting and a boundary component $F$, there is a diameter finite ball in the curve complex $\mathcal {C}(F)$ so that it contains all distance degenerate curves or slopes in $F$.

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show…

It has been shown elsewhere that a strongly irreducible Heegaard splitting surface Q of a compact orientable 3-manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we…

We prove that for any six points on the Riemann sphere there exist three disjoint closed (or open) discs, each of which contains exactly two of the six distinguished points. This statement shows that recently proposed method to numerically…

Let $M$ be a compact orientable irreducible 3-manifold and $H$ be an unstabilized genus three Heegaard splitting of $M$. In this article, we will define a simplicial complex of weak reducing pairs for $H$ and find several properties of this…

We give another proof of a theorem of Scharlemann and Tomova and of a theorem of Hartshorn. The two theorems together say the following. Let M be a compact orientable irreducible 3--manifold and P a Heegaard surface of M. Suppose Q is…

Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a…

A homeomorphism of a 3-manifold M is said to be Dehn twists on the boundary when its restriction to the boundary of M is isotopic to the identity on the complement of a collection of disjoint simple closed curves in the boundary of M. In…

We define a Heegaard-Scharlemann-Thompson (HST) splitting of a 3-manifold M to be a sequence of pairwise-disjoint, embedded surfaces, {F_i}, such that for each odd value of i, F_i is a Heegaard splitting of the submanifold of M cobounded by…

We classify isotopy classes of irreducible Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2 matrix with 0 in the lower right hand corner (as always is true when the absolute value of the…

The main result is a short effective proof of Tao Li's theorem that a closed non Haken hyperbolic 3-manifold N has at most finitely many irreducible Heegaard splittings.

We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$, by showing the deformation invariant of some open surfaces. Secondly, we show…

Let $M$ be a triangulated, oriented, connected compact $3$-manifold with connected non-empty boundary. Such a manifold admits a unique decomposition into $\triangle$-prime $3$-manifolds. In this paper, we show that the adjoint Reidemeister…

We prove the existence of a geometric characteristic submanifold for non-positively curved manifolds of any dimension greater than or equal to three. In dimension three, our result is a geometric version of the topological characteristic…

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…

When a 3-manifold admits an openbook decomposition, we get a Heegaard splitting by thickening a page. This splitting surface has a special multi curves coming from the binding. In this paper, we consider the subgroup of the Goeritz group of…

We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other…

The rectangle condition for a genus $g$ Heegaard splitting of a 3-manifold, defined by Casson and Gordon, provides a sufficient criterion for the Heegaard splitting to be strongly irreducible. However it is unknown whether there exists a…