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Small energy isotopies of loose Legendrian submanifolds

Lukas Nakamura

TL;DR

This work proves a sharp upper bound on the energy required to realize Legendrian isotopies of loose Legendrians: if two closed Legendrians possess loose charts of sizes $\varepsilon_0$ and $\varepsilon_1$, any isotopy between them can be $C^0$-approximated by a Legendrian isotopy of energy less than $\frac{\varepsilon_0}{2}+\frac{\varepsilon_1}{2}+\eta$ for any $\eta>0$. The approach combines a local Weinstein-neighbourhood energy control, Murphy's h-principle for loose Legendrians, a $C^0$-approximation of formal Legendrian embeddings by loose ones, and a fragmentation-type argument to glue small-energy pieces. As consequences, the displacement energy of a loose displaceable Legendrian is bounded by half the size of its smallest loose chart, and stabilized loose Legendrians satisfy both displacement and quantitative h-principle statements with explicit energy bounds, confirming a conjecture of Dimitroglou Rizell and Sullivan. These results provide a precise, local-to-global mechanism for achieving low-energy Legendrian isotopies in the loose category and connect topological flexibility with energetic control in contact dynamics.

Abstract

We prove that for a closed Legendrian submanifold $L$ of dimension $n \geq 2$ with a loose chart of size $η$, any Legendrian isotopy starting at $L$ can be $C^0$-approximated by a Legendrian isotopy with energy arbitrarily close to $\fracη{2}$. This in particular implies that the displacement energy of loose displaceable Legendrians is bounded by half the size of its smallest loose chart, which proves a conjecture of Dimitroglou Rizell and Sullivan.

Small energy isotopies of loose Legendrian submanifolds

TL;DR

This work proves a sharp upper bound on the energy required to realize Legendrian isotopies of loose Legendrians: if two closed Legendrians possess loose charts of sizes and , any isotopy between them can be -approximated by a Legendrian isotopy of energy less than for any . The approach combines a local Weinstein-neighbourhood energy control, Murphy's h-principle for loose Legendrians, a -approximation of formal Legendrian embeddings by loose ones, and a fragmentation-type argument to glue small-energy pieces. As consequences, the displacement energy of a loose displaceable Legendrian is bounded by half the size of its smallest loose chart, and stabilized loose Legendrians satisfy both displacement and quantitative h-principle statements with explicit energy bounds, confirming a conjecture of Dimitroglou Rizell and Sullivan. These results provide a precise, local-to-global mechanism for achieving low-energy Legendrian isotopies in the loose category and connect topological flexibility with energetic control in contact dynamics.

Abstract

We prove that for a closed Legendrian submanifold of dimension with a loose chart of size , any Legendrian isotopy starting at can be -approximated by a Legendrian isotopy with energy arbitrarily close to . This in particular implies that the displacement energy of loose displaceable Legendrians is bounded by half the size of its smallest loose chart, which proves a conjecture of Dimitroglou Rizell and Sullivan.

Paper Structure

This paper contains 5 sections, 17 theorems, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The energy of a contactomorphism
  3. Loose Legendrians
  4. Proof of Theorem \ref{['thm:main result']}
  5. Proofs of the Corollaries

Key Result

Theorem 1.1

Let $(M,\alpha)$ be either compact or equal to $(P \times \mathbb{R}, \lambda + dz)$ as above, and let $L_0, L_1$ be two distinct closed Legendrian submanifolds that can be connected by a Legendrian isotopy. If there are no Reeb chords between $L_0$ and $L_1$, thenThe additional factor of 2 in Theor

Figures (3)

  • Figure 1: After perturbing the Legendrians there exists an isotopy $\phi_t$ of small energy.
  • Figure 2: Starting from an isotopy $f_t$ of Legendrian embeddings, we construct a non-Legendrian perturbation $g_1$, a loose Legendrian approximation $\chi_0$ of $g_1$ inside of $W_0$, and contact isotopies $\Phi_t$ and $\widetilde{\phi}_t$ that satisfy $\Phi_t \circ f_0 = f_t$ and $\widetilde{\phi}_1(\widetilde{L}) = \Phi_1(\widetilde{L})$.
  • Figure 3: Inside of $V_i \cap W_i$, we find a formal Legendrian homotopy $(h^i_t,H^i_{s,t})$ from $f_i$ to a loose Legendrian embedding $h^i_1$ which admits a small loose chart.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 16 more