12 problems

Let $(Y,\xi)$ be a closed contact manifold, let $\alpha$ be any contact form on $Y$, and let $\Lambda\subset Y$ be a closed Legendrian submanifold. Arnol'd's chord conjecture. The…

Loose Legendrian rigidity conjecture. The Legendrian $\varphi_\infty(\Lambda)$ is still loose.

Legendrian Gauss-map rigidity conjecture. The diagram relating the Lagrangian Gauss maps of $\Lambda$ and $\varphi_\infty(\Lambda)$ commutes:

Let $(M,d\lambda)$ be an exact symplectic manifold, and let $L$ be a closed exact Lagrangian submanifold with vanishing Chekanov–Hofer metric. Let $\Lambda$ be the Legendrian lift…

Let $(M^{2n+1},\alpha)$ be a strict contact manifold, and let $L$ be a closed Legendrian submanifold. For Legendrians in the Legendrian isotopy class of $L$, let $d_\alpha$ denote…

Let a Legendrian be a Legendrian submanifold in the standard $3$-sphere, and let a Reeb vector field be a Reeb vector field for the contact structure. Arnol'd's Reeb chord conjectu…

Let $(M,\alpha)$ be a contact manifold and let $L$ be a closed Legendrian submanifold. A stabilization of $L$ is obtained by adding a stabilization supported in an arbitrarily smal…

Let $S^{2n-1}$ carry its standard contact structure $bxi_{std}$, let $\Lambda_s\subset (S^{2n-1},\xi_{std})$ be a Legendrian, let $D_s'$ be the corresponding object, and write…

Let $M^n$, with $n\geq2$, be a closed contact stationary Legendrian submanifold of $\mathbb{S}^{2n+1}$, and let $\mathbf{B}$ be its second fundamental form. The first-gap conjectur…

Let $M^n$, with $n\geq2$, be a closed minimal Legendrian submanifold in $\mathbb{S}^{2n+1}$, and let $\mathbf{B}$ be its second fundamental form. The minimal Legendrian pinching co…

Non-degeneracy conjecture. The $\alpha$-metric $\delta_{\alpha}$ on $\mathcal L(N)$ is non-degenerate.

Equivalence conjecture. The study of generating family-compatible Lagrangian cobordisms between Legendrian submanifolds should be tantamount to the study of zero-Maslov Lagrangian…