On horizontal immersions of discs in fat distributions of type (4,6)

In this paper, we discuss horizontal immersions of discs in certain corank-2 fat distributions on 6-dimensional manifolds. The underlying real distribution of a holomorphic contact distribution on a complex 3 manifold belongs to this class. The main result presented here says that the associated nonlinear PDE is locally invertible. Using this we prove the existence of germs of embedded horizontal discs.

Holomorphic Legendrian curves

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on$\mathbb{C}^{2n+1}$for any$n\in \mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface$M$admits a proper holomorphic Legendrian embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface$M={M\unicode[STIX]{x0030A}}\,\cup \,bM$there exists a topological embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$whose restriction to the interior is a complete holomorphic Legendrian embedding${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold$W$carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on$W$.

Generalised symmetries of partial differential equations via complex transformations

We consider two systems of real analytic partial differential equations, related by a holomorphic contact map H. We study how the generalised symmetries of the first equation are mapped into those of the second one, and determine under which conditions on H such a map is invertible. As an application of these results, an example of physical interest is discussed.