Narrow Results

In this paper, the notion of the generic knots in contact 3-manifolds is introduced as an extension of that of the transversal knots. We show that any generic knots in contact 3-manifolds are constructed by perturbations of some Legendrian knots. As for the classification problem, the situation is completely different from the cases of the transversal and Legendrian knots. Two generic knots in a contact 3-manifold are generically isotopic if and only if they have the same number of non-transversal points and belong to the same topological knot class. We treat, in this paper, not only trivial knots in tight contact 3-manifolds but also non-trivial knots and those in overtwisted contact 3-manifolds.

We describe open book decompositions of links of simple surface singularities that support the corresponding unique Milnor fillable contact structures. The open books we describe are isotopic to Milnor open books.

We express the mean Euler characteristic (MEC) of a contact structure in terms of the mean indices of closed Reeb orbits for a broad class of contact manifolds, the so-called asymptotically finite contact manifolds. We show that this class is closed under subcritical contact surgery and examine the behavior of the MEC under such surgery. To this end, we revisit the notion of index-positivity for contact forms. We also obtain an expression for the MEC in the Morse–Bott case.

A contact manifold is smoothly Milnor fillable if it is contactomorphic to the contact boundary of the Milnor fibre of a germ with no blowing up. In this note, this notion is compared with the previously defined singular Milnor fillability, both on topological and contact levels. In particular, an example where the two notions give rise to non isomorphic contact structures on the same 3-manifold is given.

In this paper, we derive a partial result related to a question of Yau: “Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?”

Main Theorem. Let M²ⁿ be a simply-connected complete Kähler manifold M with negative sectional curvature ≤−1 and S_∞(M) be the sphere at infinity of M. Then there is an explicit bounded contact form β defined on the entire manifold M²ⁿ.

Consequently, if M²ⁿ is a simply-connected Kähler manifold with negative sectional curvature −a² ≤ sec_M ≤ −1, then the sphere S_∞(M) at infinity of M admits a bounded contact structure and a bounded pseudo-Hermitian metric in the sense of Tanaka-Webster.

We also discuss several open modi_ed problems of Calabi and Yau for Alexandrov spaces and CR-manifolds.