Lagrangian Isotopy Classification

• Lagrangian isotopy classification is a framework to distinguish Lagrangian submanifolds using combinatorial invariants (like rooted forests) alongside analytical invariants such as displacement energy and Floer homology.
• The combinatorial approach encodes twist operations of monotone Lagrangian tori into rooted forests, enabling precise identification of Hamiltonian isotopy classes.
• Analytical techniques, including displacement energy germs and pearl/Floer homology computations, provide robust non-displaceability criteria that differentiate exotic tori beyond smooth isotopy.

Lagrangian isotopy classification concerns the systematic distinction and organization of Lagrangian submanifolds in symplectic manifolds up to Lagrangian isotopy, with special attention to the finer notions of Hamiltonian isotopy and symplectomorphism. Recent research has articulated a variety of classification techniques utilizing combinatorial data, analytical invariants related to energy and Floer theory, and homological methods. These provide rigid symplectic invariants capable of distinguishing even smoothly isotopic Lagrangians that are not Hamiltonian isotopic.

1. Combinatorial Classification via Rooted Forests

For certain families of monotone Lagrangian tori—specifically those constructed by twist operations in standard symplectic vector spaces, products of spheres, and complex projective spaces—the classification framework reduces to explicit combinatorial data. Each monotone twist torus, such as

$$\Theta^k = \left\{ \frac{1}{\sqrt{k+1}} (e^{i\alpha_1}\gamma(t), \ldots, e^{i\alpha_{k+1}}\gamma(t))\,\Big|\, \sum_{j=1}^{k+1}\alpha_j=0 \right\},$$

where %%%%1%%%% is an embedded sectorial plane curve, is built from a recursively defined process.

The essential combinatorial invariant here is a planar rooted forest: each primitive twist operation corresponds to a "bush" (rooted tree with $k+1$ leaves), and iteration produces forests by grafting new bushes onto previous leaves. Products of tori correspond to disjoint unions of such trees. The main result (Chekanov et al., 2010) is that two twist tori in $\times_n S^2$ are symplectomorphic and Hamiltonian isotopic if and only if their associated rooted forests are isomorphic, i.e., there is a root- and leaf-preserving homeomorphism of forests. For $\mathbb{CP}^n$, this equivalence is confirmed for $n\leq 7$, but in higher dimensions further distinctions arise.

This root-forest classification provides an efficient algebraic framework to distinguish non-standard Lagrangian tori (sometimes called exotic tori) that are not Hamiltonian isotopic to the standard product "Clifford" torus.

2. Analytical Invariants: Displacement Energy Germs and $J$-Holomorphic Discs

A second, fundamentally analytical, approach leverages displacement energy and $J$-holomorphic disc counting invariants. The displacement energy

$$e(A) = \inf\{\|H\|\:|\:\Phi_H(A)\cap A = \emptyset\}$$

quantifies the minimal Hofer energy required to separate a compact subset $A$ from itself by a Hamiltonian isotopy. For a Lagrangian torus $L$, the displacement energy germ $S^e_L:\,H^1(L;\mathbb{R})\to [0,\infty]$ is defined as $S^e_L(\xi) = e(L_\xi)$, where $L_\xi$ is the image of $L$ under a closed 1-form perturbation.

$S^e_L$ is preserved under symplectomorphisms:

$S^e_{\psi(L)} = S^e_L \circ \psi^*|_L$

for any symplectomorphism $\psi$. This functional distinguishes Hamiltonian isotopy classes of Lagrangian tori. For instance, in $S^2 \times S^2$, explicit computation yields

$S^e_{T^2}(x, y) = \min\{1 - |x|,\, 1 - |y|\}$

for the Clifford torus, whereas for a twist torus $\Theta^{1}$, one obtains

$$S^e_{\Theta}(s, t) - 1 = \min\{ s, -s + t, -s - t \}.$$

The structure of the minima (number of independent directions and linear functionals involved) encodes "rigidity": these energy-germs cannot be mapped to each other by any linear automorphism, implying the tori are not Hamiltonian isotopic.

Furthermore, the paper employs pearl or Floer homology computations for $L$: by analyzing a pearl complex whose differential combines Morse and $J$-holomorphic disc data, it proves the non-displaceability of these tori (i.e., $e(L) = \infty$), providing a robust symplectic invariant.

3. Homological and Floer-Theoretic Criteria

Non-displaceability is central to Lagrangian isotopy classification. It implies that a Lagrangian cannot be "pushed off itself" by any Hamiltonian isotopy, and Floer cohomology is nonzero. Since these homologies are invariant under Hamiltonian isotopy, the lack of isomorphism between the Floer (or pearl) homologies and the associated energy germs—demonstrated explicitly for twist versus Clifford tori—gives a method to distinguish between Hamiltonian isotopy classes, even when the Lagrangians are smoothly or even Lagrangian isotopic.

This analytic rigidity is reinforced by explicit computations of $J$-holomorphic disc contributions to the pearl complex potential function $U$ and the non-vanishing of homology in degree zero, which is sufficient to prove non-displaceability and thus isolate distinct isotopy classes.

4. Synthesis of Classification Strategies

The classification of monotone Lagrangian tori arising from twist constructions is governed by two parallel invariants:

• Combinatorial: Rooted forests encoding the inductive history of twists
• Analytical: Displacement energy germs and Floer-homological data

The interplay of these techniques yields a comprehensive classification scheme. The combinatorial rooted forest dictates the topological "type" of the torus, and the analytical displacement energy germ—computable explicitly from deformation theory and disc enumeration—verifies the robustness of the symplectic distinction. These invariants persist under symplectomorphism and Hamiltonian isotopy.

5. Applications and Broader Impact

This classification yields examples of non-displaceable, monotone Lagrangian tori in closed symplectic manifolds (e.g., products of spheres, complex projective spaces) that are not Hamiltonian isotopic to standard tori. The methods produce a rich, discrete set of monotone Lagrangian tori, whose Hamiltonian isotopy classes correspond bijectively to isomorphism classes of rooted forests. It confirms that the landscape of Lagrangian submanifolds in familiar symplectic manifolds includes an infinite zoo of rigid, "exotic" tori beyond the standard models.

The computational rigidity supplied by the energy germ and disc counting approach opens the way to extending the analysis to higher-dimensional, non-monotone, or nontrivial homology cases, as well as to investigating the relationship between combinatorial and Floer-theoretic data in more general settings.

6. Connections to Broader Symplectic Topology

These methods interact closely with current structures in symplectic topology, such as:

• Symplectic mapping class group computations via Dehn twists and their generalizations
• Mirror symmetry and the role of exotic tori in the Fukaya category
• Symplectic displacement, energy-capacity inequalities, and phenomena of rigidity versus flexibility

The explicit classification demonstrates how symplectic topology leverages both algebraic (combinatorial) and analytic (energy, Floer theory) invariants to resolve fine questions about equivalence and nonequivalence of Lagrangian submanifolds. It establishes paradigms for Lagrangian isotopy classification in settings where traditional smooth or topological invariants are insufficient, marking a transition from flexible to rigid regimes within symplectic geometry.