Sheaves via augmentations of Legendrian surfaces

The Correspondence between Augmentations and Rulings for Legendrian Knots

We strengthen the link between holomorphic and generatingfunction invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot’s contact homology to the complete ruling invariant of Chekanov and Pushkar.

Compact Special Legendrian Surfaces in S Compact Special Legendrian Surfaces in S

A surface Σ ⊂ S ⊂ C is called special Legendrian if the cone 0 × Σ ⊂ C is special Lagrangian. The purpose of this paper is to propose a general method toward constructing compact special Legendrian surfaces of high genus. It is proved there exists a compact, orientable, Hamiltonian stationary Lagrangian surface of genus 1 + k(k−3) 2 in CP 2 for each integer k ≥ 3, which is a smooth branched sur...

Rulings of Legendrian knots as spanning surfaces

Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2–graded if and only if the surface is orientable. For 2–graded rulings of homogeneous (in particular, alternating) knots, we prove that the genus of this surface is at most the genus of the knot. While this is not true in general, we do prove that the canonical genus (a.k.a. diagram genus) of any k...

Compact Special Legendrian Surfaces in S

Willmore Legendrian surfaces in pseudoconformal 5-sphere

where B0 is the trace free part of the second fundamental form of X. It was introduced by Willmore in a slightly different but equivalent form for surfaces in E. The functional is invariant under conformal transformation, and it is natural to extend it for submanifolds in conformal N -sphere S = E ∪ {∞}. Willmore conjecture asks if Clifford torus in S is the unique minimizer of W(X) among all i...