Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Let M ₁ and M ₂ be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism ${\phi : {\partial}M_1 \to {\partial}M_2}$ . We analyze the relationship between the sets of low genus Heegaard splittings of M ₁, M ₂, and M, assuming the map ${\phi}$ is “sufficiently complicated”. This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.     

Big Heegaard distance implies finite mapping class group

We show that if M is a closed three manifold with a Heegaard splitting with sufficiently big Heegaard distance then the subgroup of the mapping class group of the Heegaard surface, whose elements extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of M is finite.     

Binary relations as single primitive notions for hyperbolic three-space and the inversive plane

By interpreting J.A. Lester's [9] result on inversive-distance-preserving mappings as an axiomatizability statement, and by using the Liebmann isomorphism between the inversive plane and hyperbolic three-space, we point out that hyperbolic three-spaces (and inversive geometry) coordinatized by Euclidean fields can be axiomatized with planes (or circles) as variables, by using only the plane-orthogonality (or circle-orthogonality) predicate _p (or _c), or by using only the predicate δ′ (or δ), where δ′(p,p′) (or δ(A, B)) is interpreted as ‘the distance between the planes p and p′ is equal to the length of the segment s whose angle of parallelism is (i. e. II(s) = )’ (or as ‘the numerical distance between the disjoint circles A and B has the value , which corresponds to s via Liebmann's isomorphism’).     

Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques

We show the hyperbolic and complex function field versions of Lang’s conjecture for smooth projective surfaces S having a fibration f:S→C with orbifold base an orbifold curve of general type. The fibre multiplicities defining the orbifold base are non-classical, and simply-connected surfaces S can occur. Also, it is not possible in general to reduce to the easy case where C itself is a curve of general type by a finite étale cover c:S′→S. We leave open the arithmetic version of Lang’s conjecture, which rests on an orbifold version of Mordell’s Conjecture, consequence of the ABC-conjecture, but apparently not of the proofs of Falting’s result.     

The Rough Laplacian and Harmonicity of Hopf Vector Fields

Let M be an orientable real hypersurface of a general Kähler manifold . The characteristic vector field ξ of the induced almost contact metric structure (ξ,η, g,ϕ) is also called the Hopf vector field of M. In this paper, we compute the ‘rough’ Laplacian of ξ in terms of the shape operator A and also (as a natural generalization of the contact metric case) in terms of torsion τ = L_ξ g. Then we give some criteria of harmonicity of ξ. Moreover, we consider hypersurfaces M of contact type and give some criteria for M to admit an H-contact structure.Mathematics Subject Classifications (2000): 53C25, 53C20, 53C40, 53D35.     

Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

Let M be a 3-manifold with torus boundary components \(T_{1}\) and \(T_2\). Let \(\phi :T_{1} \rightarrow T_{2}\) be a homeomorphism, \(M_\phi \) the manifold obtained from M by gluing \(T_{1}\) to \(T_{2}\) via the map \(\phi \), and T the image of \(T_{1}\) in \(M_\phi \). We show that if \(\phi \) is “sufficiently complicated” then any incompressible or strongly irreducible surface in \(M_\phi \) can be isotoped to be disjoint from T. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.     

Covering homotopy 3-spheres

We prove the following theorem: for any closed orientable 3-manifoldM and any homotopy 3-sphere Σ, there exists a simple 3-fold branched coveringp:M→Σ. We also propose the conjecture that, for any primitive branched coveringp:M→N between orientable 3-manifolds,g(M)≥g(N), whereg denotes the Heegaard genus. By the above mentioned result, the genus 0 case of such conjecture is equivalent to the Poincaré conjecture.     

A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ?M. These minimal surfaces are either disjoint from ?M or meet ?M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ?M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.     

On the necessary and sufficient condition for the extended Wedderburn–Guttman theorem

Let A be a u by v matrix, and let M and N be u by p and v by q matrices, where p may not be equal to q or rank(M^′AN)<min(p,q). Recently, Galantai [A. Galantai, A note on the generalized rank reduction, Acta Math. Hungarica 116 (2007) 239–246] presented what he claimed to be the necessary and sufficient condition for rank(A-AN(M^′AN)^-M^′A)=rank(A)-rank(AN(M^′AN)^-M^′A) to hold. This rank subtractivity formula along with the condition under which it holds is called the extended Wedderburn–Guttman theorem. In this paper, we show that some of Galantai’s assertions are incorrect.