相关论文: Small energy isotopies of loose Legendrian submani…

We construct non-compact contact manifolds containing compact Legendrians which can be displaced from their Reeb flow with arbitrarily small oscillation energy. We use this to show the Shelukhin-Chekanov-Hofer pseudo-metric considered by…

Laudenbach and Sikorav proved that closed, half-dimensional non-Lagrangian submanifolds of symplectic manifolds are immediately displaceable as long as there is no topological obstruction. From this they deduced that the $C^0$-limit of a…

We show that Legendrian pre-quantization lifts of many non-exact Lagrangian submanifolds in $\mathbb{C}^n$ retain some quantitative rigidity from the symplectic base. In particular, they cannot be moved by Legendrian isotopy into an…

Let M be a compact symplectic manifold, and L be a closed Lagrangian submanifold which can be lifted to a Legendrian submanifold in the contactization of M. For any Legendrian deformation of L satisfying some given conditions, we get a new…

We prove that for any compact Lagrangian submanifold intersecting an open subset $U$ in tame symplectic manifold $(M,\omega)$, the Hofer displacement energy of $L$ from $U$ is positive, provided $L \cap U \neq \emptyset$. We also give an…

We give an $h$--principle type result for a class of Legendrian embeddings in contact manifolds of dimension at least $5$. These Legendrians, referred to as loose, have trivial pseudo-holomorphic invariants. We demonstrate they are…

We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $\mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently…

We apply the barcodes of persistent homology theory to the Chekanov-Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov-Eliashberg algebra to…

We show that any closed oriented immersed Hamiltonian stationary isotropic surface $\Sigma$ with genus $g_{\Sigma}$ in $S^{5}\subset\mathbb{C}^{3}$ is (1) Legendrian and minimal if $g_{\Sigma}=0$; (2) either Legendrian or with exactly…

By a classical theorem of Chekanov, the displacement energy, $e$, of a Lagrangian submanifold is bounded from below by the minimal area of pseudo-holomorphic disks with boundary on the Lagrangian, $\hbar$. We compute $e$ and $\hbar$ for…

We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has…

Loose Legendrian n-submanifolds, for n at least 2, were introduced by Murphy and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian…

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupying a bounded generalized cylinder, $ \Omega\times[0,L]$, in $\mathbb{R}^3$, where $\Omega$ is a bounded simply connected Lipschitz domain in…

Let $\mathcal{P}$ be a set of $n$ points in the Euclidean plane. We prove that, for any $\epsilon > 0$, either a single line or circle contains $n/2$ points of $\mathcal{P}$, or the number of distinct perpendicular bisectors determined by…

Consider $N$ bosons in a finite box $\Lambda= [0,L]^3\subset \bR^3$ interacting via a two-body nonnegative soft potential $V= \lambda \tilde V$ with $\tilde V$ fixed and $\lambda>0$ small. We will take the limit $L, N \to \infty$ by keeping…

We prove that the displacement energy of a stable coisotropic submanifold is bounded away from zero if the ambient symplectic manifold is closed, rational and satisfies a mild topological condition.

Let $G$ be a simple graph with $n$ vertices, $m$ edges having Laplacian eigenvalues $\mu_1, \mu_2, \dots, \mu_{n-1},\mu_n=0$. The Laplacian energy $LE(G)$ is defined as $LE(G)=\sum_{i=1}^{n}|\mu_i-\overline{d}|$, where…

It is shown that if the universal cover of a manifold $M$ is an open manifold, then two different fibres of the spherical cotangent bundle $ST^*M$ cannot be connected by a non-negative Legendrian isotopy. This result is applied to the study…

It is shown that non-negative Legendrian isotopy defines a partial order on the universal cover of the Legendrian isotopy class of the fibre of the spherical cotangent bundle of any manifold. This result is applied to Lorentz geometry in…

We show that Legendrian integral currents in a contact manifold that locally minimize the mass among Legendrian competitors have a regular set which is open and dense in their support. We apply this to show existence and partial regularity…