New constraints on Lagrangian embeddings and the shape invariant

TL;DR

This work extends Hind–Zhang’s Hamiltonian shape invariant to four dimensions by computing Sh for toric domains and establishing sharp Lagrangian rigidity/intersection results for product tori inside polydisks. The authors develop a robust toolkit of relative finite-energy foliations and neck-stretching, combined with intricate $(d,n_1)$-building analyses, to prove that certain Lagrangian tori must intersect under Hamiltonian embeddings into toric domains, and they derive explicit shape formulas for $X_{\\Omega}$ and ellipsoids. A key outcome is the exact shape computation for toric domains of the form $X_{\\Omega(f)}$ and the ellipsoid case, plus a soft embedding construction showing that intersection rigidity can vanish in packing regimes. These results illuminate the relationship between symplectic embeddings, Lagrangian intersections, and shape invariants in dimension four, with implications for rigidity phenomena and potential extensions to more general toric domains.

Abstract

For a large class of toric domains in $\mathbb{R}^4$ we determine which product Lagrangian tori can be mapped into the domain by a Hamiltonian diffeomorphism. In other words, we compute the Hamiltonian shape invariant of these toric domains, as defined by Hind and Zhang. The argument relies on new intersection results for product Lagrangian tori in symplectic polydisks. For Hamiltonian diffeomorphisms which map certain Lagrangian product tori back into the polydisk, we establish intersections between the images and a one-parameter family of product Lagrangian tori that includes (is based at) the original torus. For symplectic polydisks with area ratios less than two, we strengthen this to establish intersections between the Hamiltonian images and the original Lagrangian torus. As a soft complement to these intersection results we also present an embedding construction which demonstrates that this intersection rigidity vanishes when the one-parameter family of product Lagrangian tori is replaced by a natural packing by Lagrangian tori.

Figures (18)

• Figure 1: Any Hamiltonian image of $L(s)$ in $D(b)$ must intersect both $\bigcup_{t\in [s,b-s]}L(t)$ and $\bigsqcup_{j=1}^k\Lambda_j$.
• Figure 2: For these distinct regions $\Omega_1^+ = \Omega_2^+ =\Omega_3^+$.
• Figure 3: Equality \ref{['eq:shape']} holds for $X_{\Omega(f)}$.
• Figure 4: Equality \ref{['eq:shape']} holds when $[0,1)\times[0,1)\subset \Omega \subset [0,1) \times [0,2).$
• Figure 5: It is not known when, if ever, a Lagrangian torus $L(r,s)$ with $(r,s)$ in the red region, $\mathcal{T}(a,b)$, can be symplectically embedded into $E(a,b)$.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10