by Michael G. Sullivan
A symplectomorphism of a compact \( 2n \)-dimensional symplectic manifold \( (W, \omega) \) is a diffeomorphism \( \phi: W \rightarrow W \) which preserves the symplectic 2-form \( \phi^* \omega = \omega \). Note that it preserves the volume form \( \phi^* \omega^n = \omega^n \). Consider now a sequence of symplectomorphisms \( \phi_j \) which converges in the \( C^0 \)-topology (that is, uniformly) to a diffeomorphism \( \psi: W \rightarrow W \). A measure-theory argument readily shows that \( \psi \) is volume-preserving; however, since \( C^0 \)-convergence says nothing about the sequence of derivative maps \( d \phi_j \), it is unclear if \( \psi \) is a symplectomorphism. Gromov’s Alternative in some sense is that the \( C^0 \)-closure is “all or nothing” ([e3], Section 3.4.4). More specifically, if the \( C^0 \)-closure of the subgroup of Hamiltonian diffeomorphisms (see Section 1) contains a diffeomorphism \( \psi \) which is neither symplectic nor antisymplectic (\( \psi^*\omega \ne \pm \omega) \), then (assuming either \( H^1(W) = 0 \) or certain weaker conditions), the closure contains the volume-preserving diffeomorphisms. The Eliashberg–Gromov theorem picks which alternative.
This short note describes Eliashberg’s role in this, one of the earliest and most celebrated results of symplectic rigidity, and some subsequent developments by others since his involvement/creation of this line of research. The note is far from comprehensive, as many other researchers have since developed a wide variety of related questions, tools and answers. In particular, Section 2 surveys Eliashberg’s approach while Section 3 discusses the better-known version which relies on pseudoholomorphic curves and capacities, and some subsequent related results. We begin in Section 1 with some introductory concepts.
1. Some introductory concepts
The foundational example of a symplectic manifold is the cotangent bundle \( W = T^*X \) of a smooth \( n \)-manifold, with symplectic form \( \omega =-d \lambda \), where \( \lambda \) is the tautological one-form. A special case is the standard symplectic structure on \( T^*\mathbb{R}^{2n} = \mathbb{C}^n \) (resp. any open subset) with \( \omega = \sum_{j=1}^n \,dx_j\, dy_j \) (resp. \( \omega \) restricted to the subset).
Pick a (compactly supported) time-dependent \( H_t \in C^\infty(W, \mathbb{R}) \), define the vector field \( X_{H_t} \) by \[ \omega(X_{H_t}, \cdot\,) = dH_t, \] then define the isotopy \( \phi^H_t: W \rightarrow W \) by \[ \frac{d}{dt} \phi^H_t = X_{H_t} \circ \phi^H_t\quad \text{and}\quad \psi^H_0 = \operatorname{id}. \] We call \( \phi^H_1 \) a Hamiltonian diffeomorphism and it satisfies \( (\phi^H_1)^* \omega = \omega \).
A submanifold \( L \) of a \( 2n \)-dimensional symplectic manifold \( (W, \omega) \) is Lagrangian if \( \dim(L) = n \) and \( \omega|_{TL} = 0 \). For example, \( (S^1(r))^n \subset \mathbb{C}^n \) is the standard Lagrangian torus with radii \( r \).
A contact manifold \( (M, \xi) \) is a \( (2n+1) \)-dimensional manifold along with a \( 2n \)-dimensional maximally nonintegrable distribution \( \xi \subset TM \).
A submanifold \( \Lambda \subset M \) is Legendrian if \( \dim(\Lambda) = n \) and \( T\Lambda \subset \xi \). For example, choose a (smooth) \( n \)-manifold \( X \) and \( S \in \{\mathbb{R}^1_z, S^1_z(r)\} \), then the one-jet space \( J^1(X, S) : = T^*X \times S \) is equipped with the standard contact structure \( \xi = \ker\{dz - \lambda\} \). Under the front projection \( \pi_F: J^1(X, S) \rightarrow J^0(X):= X \times S \), \( \pi_F(\Lambda) \) is a wavefront locally given by multiple graphs of locally defined functions \( z = f(x_1, \dots, x_n) \) meeting together at cusp (and higher-order) wavefront singularities. The Legendrian condition is equivalent to recovering the projected coordinates by \( y_j = \partial f/\partial x_j \).
2. Eliashberg’s approach
Gromov’s Nonsqueezing Theorem 3.1, with its celebrated proof of compactifying the moduli spaces of pseudoholomorphic curves [e2], is often cited as the beginning of modern-day symplectic geometry. What is less well known is that Eliashberg had at the same time independently sketched a similar nonsqueezing theorem using completely different techniques. Eliashberg announced it in [1] and then proved it modulo the proof of a combinatorial structure theorem for Legendrian fronts ([2], Theorem 1.8). He then showed that his version of nonsqueezing implied Theorem A. After the appearance of Gromov’s pseudoholomorphic curves method, Eliashberg never finished the combinatorial proof. This section sketches Eliashberg’s approach.
Proof. Here is a quick summary of ([2], pp. 228–230) using much of its notation.
Let \( B = \mathbb{C} \times (\mathbb{C} \setminus 0)^{n-1} \). Fix \( \theta \in S^1 \) and define for any embedded Lagrangian torus \( g: (S^1)^n \rightarrow B \) the “first-factor period” \[ P(g):= \int_{S^1} (g|_{S^1 \times \theta \times \dots \times \theta})^* \sum_j x_j\, dy_j. \] Let \( \mathcal{L} \) be the space of Lagrangian embedded tori in \( B \) with positive first-factor period, and let \( \mathcal{L}_0 \subset \mathcal{L} \) be the component containing the standard tori \( (S^1(r))^n \) from Section 1. Eliashberg shows, using Lees’ theory of Lagrangian immersions [e1], that \( f|_{\partial D_a \times (\partial D_R)^{n-1}} \in \mathcal{L}_0 \), where \( f \) is from the theorem, and that the theorem follows upon verifying \( P(g) \le 1 \) for any \( g \in \mathcal{L}_0 \) satisfying \( g((S^1)^n) \subset \{ 0 \le x_1, y_1 \le 1\} \times (\mathbb{C} \setminus 0)^{n-1} \).
Consider an isotopy \( g_t \in \mathcal{L}_0 \) such that \( g_0 \) is any \( g \in \mathcal{L}_0 \) and \( g_1((S^1)^n)= (S^1(r))^n \subset B \) for some \( r \). After rescaling, assume \( P(g_t) = 2\pi r \) for all \( t \). Consider the symplectic exponential covering map from \( B_+:=\mathbb{C} \times (\{z \in \mathbb{C}\,\,| \,\,\mbox{Im}z > 0\})^{n-1} \) to \( B \), under which it is easy to lift \( g_t \subset B \) to a Lagrangian isotopy \( g^+_t \subset B_+ \). Since \( P(g_t) = 2\pi r \), \( g^+_t \) further lifts to a Legendrian isotopy \( \widetilde{g}_t: (S^1)^n \rightarrow J^1(\mathbb{R}^n, S^1(r)) \) when viewing \( B_+ \subset \mathbb{C}^n = T^*\mathbb{R}^n \).
Consider the fiber bundle \[ \mathbb{R}^n = \mathbb{R}^1_{x_1} \times \mathbb{R}^{n-1}_{x_2, \dots, x_n} \rightarrow \mathbb{R}^{n-1}_{x_2, \dots, x_n} .\] For each \( t \), the front \( \pi_F(g_t((S^1)^n)) \) over a given \( (x_2, \dots, x_n) \) is the front of a 1-dimensional Legendrian. Eliashberg introduces a “decomposable 1-dimensional front”, which is roughly a union of paths oriented with respect to the base \( \mathbb{R}^1_{x_1} \), combined with non-self-intersecting cycles each with exactly 2 cusps. He then states (unfortunately without proof) that the above 1-dimensional fronts, which make up \( \pi_F(g_t((S^1)^n)) \), are decomposable ([2], Theorem 1.8). A quick computation of the first-factor period for each of the cycles and paths in the decomposition finishes the proof. ◻
Using this specific nonsqueezing Theorem 2.1, Eliashberg then provides a short proof of Theorem A ([2], p. 230). Although he does not say it, he essentially is using symplectic capacities as defined in Section 3. The explicit connection to symplectic capacities was done by Ekeland and Hofer [e4].
Eliashberg’s first step in the proof was to note that if \( A: V \rightarrow V \) is a linear isomorphism on a symplectic vector space \( (V, \omega) \), then either \( A^* \omega = \pm \omega \), or there exists a symplectic basis such that \[A = \left( \begin{array}{c|c} \begin{matrix} \lambda & 0\\ 0 & \lambda \end{matrix} & \mathbf{0} \\ \hline * & * \end{array}\right) \] for some \( 0 < \lambda < 1 \). This is first stated as ([2], Lemma 2.3.2) with details provided later in ([e9], pp. 60–61).
Consider a sequence of symplectomorphisms \( \phi_n \) on \( (W,\omega) \) which \( C^0 \)-converges to the diffeomorphism \( \psi \). Darboux’s Theorem states locally \( (W, \omega) \) looks like standard \( \bigl(\mathbb{R}^{2n}, \sum_{j=1}^n dx_j \,dy_j\bigr) \). So it suffices to show that for \( \mathbf{0} \in \mathbb{R}^{2n} \), \( \,A:= d\phi_{\mathbf{0}} \) is a symplectic linear map. Suppose not. Since \( \phi \) is volume-preserving, \( A \) is an isomorphism. A simple stabilization trick reduces the antisymplectic to the symplectic case. So the above matrix decomposition applies to \( A \). For suitable \( R_2, \dots, R_{n} \), \( A(\partial D_1 \times \partial D_{R_2} \times \dots \times \partial D_{R_n}) \subset B \) and \( A|_{\partial D_1 \times \partial D_{R_2} \times \dots \times \partial D_{R_n}} \) is homotopic to the inclusion. \( C^0 \)-convergence implies there exists \( \phi_{n} \) (for large enough \( n \)) and \( \tilde{\lambda} < 1 \) such that \[ \phi_{n}(\partial D_1 \times \partial D_{R_2} \times \dots \times \partial D_{R_n}) \subset D_{\tilde{\lambda}} \times (\mathbb{C}\setminus 0)^{n-1} \] with the restriction also homotopic to the inclusion. This contradicts Theorem 2.1 with \( a:= \tilde{\lambda} \) and \( b:=1 \).
3. Other people
As alluded to in Section 2, Eliashberg did not complete the analysis of \( n \)-dimensional Legendrian fronts using 1-dimensional Legendrian slices once Gromov had proved the following nonsqueezing theorem using pseudoholomorphic curves.
Ekeland and Hofer then defined symplectic capacities based on Gromov’s result [e4]. One version is the following.
- If \( (W_1, \omega_1) \) symplectically embeds into \( (W_2, \omega_2) \) and \( \dim W_1 = \dim W_2 \), then \( c(W_1, \omega_1) \le c(W_2, \omega_2) \).
- \( c(W, \lambda \omega) = \lambda c(W, \omega) \).
- For the canonical symplectic structure, \( c(D_1^{2n}) = \pi = c(D_1^2 \times \mathbb{C}^{n-1}) \).
The nonobvious fact is the existence of a symplectic capacity. Indeed, Theorem 3.1 is equivalent to the existence of a capacity. That existence implies nonsqueezing follows from item (1). Nonsqueezing implies that the so-called Gromov width \( w \) is a capacity, where \[ w(W, \omega):= \sup\{\pi r^2\,|\,D^{2n}_r \,\,\text{symplectically embeds in } W\}. \] McDuff and Salamon offer a nice survey of how to explicitly use capacities to prove that Theorem 3.1 implies Theorem A ([e8], Section 12.2). The key step there is to show that a diffeomorphism on \( \mathbb{R}^{2n} \) is a symplectomorphism or antisymplectomorphism if and only if it preserves some capacity for all symplectic ellipsoids.
There is a third important ingredient closely related to capacities and nonsqueezing, which is Hofer’s geometry on the space of Hamiltonian diffeomorphisms. Two Hamiltonian diffeomorphisms \( \phi \) and \( \phi^{\prime} \) can be connected by a Hamiltonian isotopy \( \phi_t^H \) for some time-dependent Hamiltonian \( H_t \in C^\infty(W, \mathbb{R}) \). Define \[ \rho(\phi, \phi^{\prime}) = \inf_{\{\phi_t^H\,\,|\,\, \phi_0^H = \phi, \phi_1^H = \phi^{\prime}\}} \int_0^1 (\sup_{x \in W} H_t(x) - \inf_{x \in W} H_t(x))\, dt. \] This is a biinvariant (under composition by Hamiltonian diffeomorphism) pseudonorm, known as Hofer’s norm. Hofer, Polterovich and then Lalonde–McDuff [e5], [e6], [e7] proved that this is in fact a norm. As mentioned in ([e7], Remark 2.3), this (hard) nondegeneracy is equivalent to the generalization of the nonsqueezing theorem proved in [e7].
The crucial ingredients in the above works, and many related ideas, such as Gromov width, displacement energy, Hofer norm, spectral norm (generalized to barcodes) and the Hofer–Zehnder capacity, were later developed and adapted to Lagrangian submanifolds, contact manifolds and Legendrian submanifolds. The contact and Legendrian versions of this story are currently less complete. All of these form an extremely active area of research, so it might be improper to start citing some mathematicians and not others. For the most part, the researchers call this area \( C^0 \)-symplectic topology, no doubt after \( C^0 \)-closure as first proved in the Eliashberg–Gromov theorem.
