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Recovering contact forms from boundary data

Gabriel Katz

TL;DR

This work develops boundary-to-bulk holography for contact manifolds by linking boundary data of traversing Reeb flows to the bulk contact geometry. The core achievement is the Holography Theorem, which shows that boundary information encoded in the causality map $C_{v_eta}$, the boundary contact form $eta^ ext{∂}$, and the boundary Lyapunov data $(f^ullet)^ ext{∂}$ suffices to reconstruct $(X,f^ullet,eta)$ up to a boundary-fixing diffeomorphism, and, under further data, to recover $eta$ exactly. The paper also develops a non-squeezing-type framework for contact embeddings, introduces Morse-wrinkle invariants $oldsymbol{ ho}_j(eta)$ and $oldsymbol{ ho}_j^+(eta)$ to quantify boundary wrinkling, and proves their invariance under certain boundary-deformations; it further links Legendrian submanifolds to boundary shadows and provides reconstruction procedures for these submanifolds from their $(-v_eta)$-shadows. Collectively, these results advance inverse-boundary problems in contact geometry, offering analytic and geometric tools to recover interior dynamics and structures from boundary data with potential applications in geometric analysis and inverse problems.

Abstract

Let $X$ be a compact connected 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.

Recovering contact forms from boundary data

TL;DR

This work develops boundary-to-bulk holography for contact manifolds by linking boundary data of traversing Reeb flows to the bulk contact geometry. The core achievement is the Holography Theorem, which shows that boundary information encoded in the causality map , the boundary contact form , and the boundary Lyapunov data suffices to reconstruct up to a boundary-fixing diffeomorphism, and, under further data, to recover exactly. The paper also develops a non-squeezing-type framework for contact embeddings, introduces Morse-wrinkle invariants and to quantify boundary wrinkling, and proves their invariance under certain boundary-deformations; it further links Legendrian submanifolds to boundary shadows and provides reconstruction procedures for these submanifolds from their -shadows. Collectively, these results advance inverse-boundary problems in contact geometry, offering analytic and geometric tools to recover interior dynamics and structures from boundary data with potential applications in geometric analysis and inverse problems.

Abstract

Let be a compact connected smooth manifold with boundary. The paper deals with contact -forms on , whose Reeb vector fields admit Lyapunov functions . We tackle the question: how to recover and from the appropriate data along the boundary ? We describe such boundary data and prove that they allow for a reconstruction of the pair , up to a diffeomorphism of . We use the term ``holography" for the reconstruction. We say that objects or structures inside are {\it holographic}, if they can be reconstructed from their -flow induced ``shadows" on the boundary . We also introduce numerical invariants that measure how ``wrinkled" the boundary is with respect to the -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.
Paper Structure (9 sections, 41 theorems, 123 equations, 3 figures)

This paper contains 9 sections, 41 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Traversing, boundary generic, & traversally generic vector fields
  3. Holography on manifolds with boundary and causality maps
  4. Contact Holography: recovering contact geometry from the boundary data
  5. Boundary data of scattering maps $C_{v_\beta}$ and the group-theoretical sections of $\beta$-contactomorphisms
  6. On contact forms and contact vector fields on manifolds with boundary
  7. Non-squeezing of $2n$-symplectic volumes, Morse wrinkles of Reeb fields, and holography
  8. Invariance of the volumes $\{\kappa_j(\beta)\}$ under special deformations of $\beta$
  9. Shadows of Legandrian submanifolds of $(X, \beta)$ on the screen $\partial_1^+X(v_\beta)$

Key Result

Lemma 4.1

Let $v$ be a smooth traversing vector field and $\beta$ a $1$-form on a compact manifold $X$ such that $\beta(v) = 1$. Then $v$ admits a Lyapunov function $f: X \to \mathbb R$ such that $df(v) = 1$. Moreover, for any $v$-trajectory $\gamma$ and any closed interval $[a, b] \subset \gamma$, we have $\

Figures (3)

  • Figure 1: The figure helps with the notations in the proof of Theorem \ref{['th.non-squeezing']}. In the figure, $v_{\beta_1}$ and $v_{\beta_2}$ are the constant vertical vector fields and $c^\bullet(\Psi, v_{\beta_2}) = 2$.
  • Figure 2: By Theorem \ref{['th.detecting_volumes']}, $\kappa_1^+(\beta_X) = \kappa_2(\beta_X) = -\int_{\mathcal{Y}_{1}^+}d\beta_Y$. The figure illustrates the geometry of $3$-folds $X$ and $Y$ and of an embedding $\Psi: X \to Y$ that leads to this formula. Although the relevant for the formula loci are: the surface $\Psi(\partial_1^+X)$, the solid $\mathcal{X}^+_1 \subset Y$, and the surface $\mathcal{Y}^+_1 \subset \partial_1^+ Y$, they are not shown in the figure due to artistic limitations; instead, we have chosen to show similar loci of one dimension lower: the curve $\Psi(\partial_2^+X)$, the surface $\mathcal{X}^+_2$, and the curve $\mathcal{Y}^+_2$.
  • Figure 3: The $(-v_\beta)$-shadow $L^\dagger \subset \partial_1^+X(v_\beta)$ of a closed Legandrian manifold $L \subset X$, shown as a bold loop in the interior of the bulk $X$. In the figure, $L^\dagger$ consists of two arcs. The surface $L(v_\beta)$ consists of $(-v_\beta)$-trajectories falling down from $L$. Note the locus $\delta L(v_\beta) \subset X$ that consists of $(-v_\beta)$-trajectories falling down from $L^\dagger \cap \partial_2^+X(v_\beta)$.

Theorems & Definitions (104)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Lemma 4.1
  • proof
  • Corollary 4.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • ...and 94 more