Recovering contact forms from boundary data

TL;DR

This paper demonstrates how to reconstruct a compact manifold with boundary and a contact form from boundary data, introducing holographic principles and invariants, with results analogous to symplectic non-squeezing.

Contribution

It establishes a method for recovering the manifold and contact form from boundary data and introduces numerical invariants measuring boundary complexity.

Findings

Reconstruction of manifold and contact form from boundary data.

Introduction of boundary complexity invariants.

Non-squeezing results for contact embeddings.

Abstract

Let $X$ be a compact smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\beta$ admit Lyapunov functions $f$. We tackle the question: how to recover $X$ and $\beta$ 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 $(X, \beta)$, 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 $v_\beta$-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 $v_\beta$-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.