Recovering contact forms from boundary data
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911729747755008 |
|---|---|
| author | Katz, Gabriel |
| author_facet | Katz, Gabriel |
| contents | Let $X$ be a compact smooth manifold with boundary. The paper deals with contact $1$-forms $β$ on $X$, whose Reeb vector fields $v_β$ admit Lyapunov functions $f$.
We tackle the question: how to recover $X$ and $β$ from the appropriate data along the boundary $\partial X$? We describe such boundary data and prove that they allow for a reconstruction of the pair $(X, β)$, up to a diffeomorphism of $X$. We use the term ``holography" for the reconstruction. We say that objects or structures inside $X$ are {\it holographic}, if they can be reconstructed from their $v_β$-flow induced ``shadows" on the boundary $\partial X$.
We also introduce numerical invariants that measure how ``wrinkled" the boundary $\partial X$ is with respect to the $v_β$-flow and study their holographic properties under the contact forms preserving embeddings of equidimensional contact manifolds with boundary. We get some ``non-squeezing results" about such contact embedding, which are reminiscent of Gromov's non-squeezing theorem in symplectic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14604 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Recovering contact forms from boundary data Katz, Gabriel Symplectic Geometry Geometric Topology 53D10, 53D12 Let $X$ be a compact smooth manifold with boundary. The paper deals with contact $1$-forms $β$ on $X$, whose Reeb vector fields $v_β$ admit Lyapunov functions $f$. We tackle the question: how to recover $X$ and $β$ from the appropriate data along the boundary $\partial X$? We describe such boundary data and prove that they allow for a reconstruction of the pair $(X, β)$, up to a diffeomorphism of $X$. We use the term ``holography" for the reconstruction. We say that objects or structures inside $X$ are {\it holographic}, if they can be reconstructed from their $v_β$-flow induced ``shadows" on the boundary $\partial X$. We also introduce numerical invariants that measure how ``wrinkled" the boundary $\partial X$ is with respect to the $v_β$-flow and study their holographic properties under the contact forms preserving embeddings of equidimensional contact manifolds with boundary. We get some ``non-squeezing results" about such contact embedding, which are reminiscent of Gromov's non-squeezing theorem in symplectic geometry. |
| title | Recovering contact forms from boundary data |
| topic | Symplectic Geometry Geometric Topology 53D10, 53D12 |
| url | https://arxiv.org/abs/2309.14604 |