Moduli of Legendrian foliations and quadratic differentials in the Heisenberg group

Forme semi-locale des feuilletages legendriens

We describe a semi-local canonical form for Legendrian foliations on contact manifolds in the neighbourhood of a Legendrian submanifold. This result generalizes local results by Libermann and Pang on Legendrian foliations on contact manifolds, and is analogeous to a semi-local result by Weinstein in the symplectic case. For the proof, we introduce and use a class of cohomology that obstructs the possibility to make a Legendrian foliation flat.

Legendrian Gronwall conjecture

The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjecture states that a Legendrian 3-web admits at most one distinct local linearization, with the only exception when it is locally equivalent to the dual linear Legendrian 3-web of the Legendrian twisted cubic in $\,\PP^3$. We give a partial answer to the conjecture in the affirmative …

Almost Monotonicity Formula for H-minimal Legendrian Surfaces in the Heisenberg Group

We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467

Legendrian Submanifold Path Geometry

Let $Z \to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is \[ Sp(n+1, R) \to RP^{2n+1}. \] The equivalence problem provides an $sp(n+1, R)$ valued Cartan connection form that captures the geometry of such foliations. Two special cases are considered. The first case is characterized by having a well defined conformal class of symmetric $(n+1)$ differentials on the space of leaves of the foliation $X$. The $G$ structur…

Legendrian 纽结在海森堡群中的 CR 不变能量

CR-invariant energy of Legendrian knots in the Heisenberg group

本文研究了海森堡群中 Legendrian 纽结的 CR 不变能量，通过正则化 Korányi 距离下的 -2 阶势能积分来定义能量函数，并展示了实数圆作为能量最小化者的特性，同时给出了海森堡群中的 Doyle--Schramm 余弦公式类比。

Bi-paracontact structures and Legendre foliations

We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold $(M,η)$, then under some natural assumptions of integrability, $M$ carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then $M$ admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of an anti-hypercomplex (or complex-pro…