05.07.04 · symplectic / gromov

Eliashberg-Gromov $C^{0}$ -rigidity of $Symp$

shipped3 tiersLean: none

Anchor (Master): Eliashberg 1981 *Funct. Anal. Appl.* 15 (announcement); Gromov 1985 *Invent. Math.* 82; Hofer 1990 *Proc. R. Soc. Edinburgh* 115A; Ekeland-Hofer 1989 *Math. Z.* 200; Müller-Oh 2007 *J. Symplectic Geom.* 5; Buhovsky 2008 *J. Symplectic Geom.* 6; Humilière-Leclercq-Seyfaddini 2015; Buhovsky-Opshtein 2016

Intuition Beginner

Symplectic geometry is defined by a derivative condition: a map $ϕ$ is a symplectomorphism if it pulls back the symplectic form $ω$ to itself, $ϕ^{*} ω = ω$ . Pulling back a form involves differentiating $ϕ$ , so the natural setting for the definition is $C^{1}$ maps — maps whose first derivatives are continuous. Naively, the condition should detect $C^{1}$ data and break down under coarser convergence.

The Eliashberg-Gromov rigidity theorem says it does not. If a sequence of symplectomorphisms $ϕ_{n}$ converges uniformly (in the $C^{0}$ topology — the topology of uniform convergence, without any derivative control) to a smooth diffeomorphism $ϕ$ , then $ϕ$ is automatically a symplectomorphism. The condition that should require $C^{1}$ control is preserved under $C^{0}$ limits.

This is striking. Volume-preserving diffeomorphisms have no such property: a $C^{0}$ -limit of volume-preserving smooth maps can fail to be volume-preserving, because oscillations on small scales can hide volume distortion. Symplectic maps are rigid in a way that volume-preserving maps are not, and the proof uses the same pseudoholomorphic-curve technology that produced Gromov non-squeezing.

Visual Beginner

A schematic illustration of two squeezing problems. A unit ball is mapped by a sequence of symplectomorphisms into a thin slab; the squeezing radius is measured by the symplectic capacity, which equals $π$ for both ball and the obstructing cylinder. Below, the same sequence converges $C^{0}$ -uniformly to a limit map; the limit's behaviour on the same ball must respect the same capacity bound, forcing it to be symplectic too.

The key picture is the dialogue between $C^{0}$ -convergence and a $C^{0}$ -continuous numerical invariant: capacity does not see derivatives, yet it forces the limit to satisfy a derivative condition.

Worked example Beginner

Take the plane $R^{2}$ with the area form $ω_{0} = d x \land d y$ , the canonical symplectic form in dimension two. In dimension two, "symplectic" means "area-preserving and orientation-preserving" — a 2-form is the only non-degenerate form there is, and any orientation-preserving area-preserving diffeomorphism is a symplectomorphism.

Take the sequence of linear maps $A_{n} : R^{2} \to R^{2}$ given in coordinates by

A_{n} (x, y) = (n \cdot x, \frac{1}{n} \cdot y) .

Each $A_{n}$ has determinant $1$ , so each is area-preserving, so each is a symplectomorphism. Now look at the limit. The sequence does not converge in the $C^{0}$ topology — the first coordinate blows up. So this sequence is not a counterexample to rigidity; it is an example of a sequence of symplectomorphisms whose $C^{0}$ -limit simply does not exist.

Now look at a different sequence: $B_{n} (x, y) = (x, y + \frac{1}{n} sin (n x))$ . Each $B_{n}$ has Jacobian determinant $1$ (compute: $1 \cdot 1 - 0 \cdot cos (n x) = 1$ ), so each is area-preserving. The sequence converges in $C^{0}$ to the identity $ϕ (x, y) = (x, y)$ , which is symplectic. Notice that the derivatives of $B_{n}$ with respect to $x$ contain $cos (n x)$ , which oscillates wildly as $n$ grows — the convergence is genuinely $C^{0}$ , not $C^{1}$ . The limit is symplectic anyway.

What this tells us: the rigidity theorem says this is the general pattern. Whenever a $C^{0}$ -limit of symplectomorphisms is itself a diffeomorphism, the limit is symplectic. The proof in dimension two is easy because area is itself $C^{0}$ -continuous; the deep content of Eliashberg-Gromov is the higher-dimensional version, where symplectic is more than area-preserving.

Check your understanding Beginner

Exercise (easy, multiple choice).

What does it mean for a sequence of diffeomorphisms $ϕ_{n}$ to converge to $ϕ$ in the $C^{0}$ topology on a compact manifold?

A. $ϕ_{n}$ and all their derivatives converge uniformly to $ϕ$ and its derivatives
B. $ϕ_{n}$ converge uniformly to $ϕ$ on every compact set, with no derivative control
C. $ϕ_{n} (x) \to ϕ (x)$ for every point $x$ separately, with no uniformity
D. $ϕ_{n}$ converge in $L^{2}$ to $ϕ$

Hint

The $C^{0}$ topology is the topology of uniform convergence — values converge uniformly, but nothing about derivatives.

Answer

B. $ϕ_{n}$ converge uniformly to $ϕ$ on every compact set, with no derivative control. The $C^{0}$ topology cares only about function values, not about derivatives. Option A describes $C^{\infty}$ convergence; option C is pointwise convergence (strictly weaker than uniform); option D is integral convergence (incomparable to uniform).

Exercise (easy, multiple choice).

The Gromov symplectic capacity $c_{G}$ of an open set $U \subset R^{2 n}$ is defined as:

A. The volume of $U$ divided by $π^{n}$
B. $π r^{2}$ , where $r$ is the largest $r$ such that $B^{2 n} (r)$ symplectically embeds into $U$
C. The diameter of $U$ in the Euclidean metric
D. The number of points where a generic Hamiltonian flow has fixed points

Hint

The Gromov width is built from non-squeezing: it asks how big a ball you can squeeze into a region without breaking the symplectic structure.

Answer

B. $π r^{2}$ , where $r$ is the largest $r$ such that $B^{2 n} (r)$ symplectically embeds into $U$ . This is the Gromov width (or Gromov symplectic capacity), the simplest of the symplectic capacities. By Gromov non-squeezing, the same number is the capacity of any cylinder $Z^{2 n} (r) = B^{2} (r) \times R^{2 n - 2}$ with cross-section radius $r$ .

Formal definition Intermediate+

Let $(M, ω)$ be a smooth symplectic manifold of dimension $2 n$ , and write $Symp (M, ω) \subset Diff (M)$ for the subgroup of $ω$ -preserving diffeomorphisms. Equip $Diff (M)$ with the $C^{0}$ -topology: the topology generated by the basic open sets

U (ϕ, K, ε) = {ψ \in Diff (M) : d_{M} (ϕ (x), ψ (x)) < ε for all x \in K},

where $K \subset M$ is compact, $ε > 0$ , and $d_{M}$ is the distance function of any auxiliary Riemannian metric on $M$ . The topology is independent of the chosen metric. For non-compact $M$ the $C^{0}$ -topology is the topology of uniform convergence on compact subsets in both $ϕ$ and $ϕ^{- 1}$ , in line with the convention adopted in Hofer-Zehnder ^{[Hofer-Zehnder]} and McDuff-Salamon ^{[McDuff-Salamon]}.

Theorem (Eliashberg-Gromov $C^{0}$ -rigidity, 1981-1985). The subgroup $Symp (M, ω)$ is closed in $Diff (M)$ with respect to the $C^{0}$ -topology. Equivalently, if $ϕ_{n} \in Symp (M, ω)$ converges to $ϕ \in Diff (M)$ in $C^{0}$ , then $ϕ \in Symp (M, ω)$ — that is, $\phi^\omega = \omega$.*

Two formulations of the theorem coexist in the literature. The smooth version assumes the limit $ϕ$ is a smooth diffeomorphism and concludes $ϕ^{*} ω = ω$ . The homeomorphism version, due to Müller-Oh 2007 The group of Hamiltonian homeomorphisms and $C^{0}$ -symplectic topology ^{[Müller-Oh 2007]}, drops the smoothness assumption: it defines a symplectic homeomorphism of $M$ as a homeomorphism that is the $C^{0}$ -limit of a sequence of symplectomorphisms, and the resulting closure

Sympeo (M, ω) := \overline{Symp (M, ω)}^{C^{0}} \subset Homeo (M)

is the natural setting for $C^{0}$ -symplectic topology. The Eliashberg-Gromov theorem in the smooth form says that the inclusion $Sympeo (M, ω) \cap Diff (M) = Symp (M, ω)$ is an equality, not a strict inclusion.

For the sign convention, this unit uses

ω (X_{H}, -) = d H,

matching 05.07.01 and 05.07.02.

A symplectic capacity (in the sense of 05.07.02) is a function $c$ assigning to each open subset $U \subset M$ a number $c (U) \in [0, \infty]$ such that monotonicity ( $U \subset V \Rightarrow c (U) \leq c (V)$ ), symplectic invariance ( $c (ϕ (U)) = c (U)$ for every $ϕ \in Symp (M, ω)$ defined on $U$ ), and the normalisation $c (B^{2 n} (r)) = c (Z^{2 n} (r)) = π r^{2}$ all hold, where $B^{2 n} (r)$ is the open Euclidean ball of radius $r$ and $Z^{2 n} (r) = B^{2} (r) \times R^{2 n - 2}$ is the open cylinder.

Counterexamples to common slips:

The hypothesis " $ϕ$ is a diffeomorphism" matters. A $C^{0}$ -limit of symplectomorphisms can be a homeomorphism that is not differentiable; the conclusion $ϕ^{*} ω = ω$ then has no naive sense, and the Müller-Oh definition replaces it by membership in $Sympeo$ . Without the smoothness hypothesis the smooth-form statement is vacuous.
The convergence must be $C^{0}$ , not pointwise. A pointwise limit of symplectomorphisms can fail to be a diffeomorphism altogether, and the theorem makes no claim in that setting.
The theorem is not a statement about $C^{1}$ -rigidity at the level of derivative tensors. The map $D ϕ_{n}$ does not converge to $D ϕ$ in any reasonable sense — the proof bypasses derivatives entirely by routing through $C^{0}$ -continuous functionals (capacities).

Key theorem with proof Intermediate+

Theorem (Eliashberg-Gromov, capacity-based proof). Let $(M, ω)$ be a symplectic manifold, $ϕ_{n} \in Symp (M, ω)$ a sequence converging $C^{0}$ -uniformly on compact subsets to a smooth diffeomorphism $ϕ \in Diff (M)$ . Then $\phi^\omega = \omega$.*

Proof. Suppose for contradiction that $ϕ^{*} ω \neq = ω$ . Then there is a point $p_{0} \in M$ where $ϕ^{*} ω - ω$ is a non-zero 2-form. By picking Darboux coordinates centred at $p_{0}$ on a neighbourhood, we may assume $M = R^{2 n}$ with $ω = ω_{0} = \sum_{i} d q_{i} \land d p_{i}$ and $p_{0} = 0$ , and we may compose $ϕ$ with the inverse of its derivative at $0$ to assume $D ϕ (0) = id$ . The hypothesis $ϕ^{*} ω \neq = ω$ at $0$ then becomes the statement that the Taylor expansion of $ϕ$ at $0$ deviates from the identity at second order in a way that produces a non-zero $ϕ^{*} ω - ω$ near $0$ .

The argument uses the Gromov symplectic capacity $c_{G}$ defined in 05.07.02: for an open set $U \subset R^{2 n}$ ,

c_{G} (U) := sup {π r^{2} : B^{2 n} (r) ↪_{symp} U},

where $↪_{symp}$ denotes symplectic embedding. By non-squeezing 05.07.01, $c_{G}$ satisfies the normalisation $c_{G} (B^{2 n} (r)) = c_{G} (Z^{2 n} (r)) = π r^{2}$ and is monotone under symplectic embeddings.

Step 1: $c_{G}$ is continuous under $C^{0}$ -convergence. Let $V \subset M$ be a precompact open set. We claim $c_{G} (ϕ_{n} (V)) \to c_{G} (ϕ (V))$ . The proof uses two ingredients: (a) for any $ε > 0$ , if $n$ is large enough then $ϕ_{n} (V) \subset (ϕ (V))^{+ ε}$ (the $ε$ -neighbourhood of $ϕ (V)$ in the auxiliary metric) and conversely $ϕ (V) \subset (ϕ_{n} (V))^{+ ε}$ , both following from uniform $C^{0}$ -closeness; (b) the Gromov width is upper-semicontinuous under monotone limits of open sets, $c_{G} (U^{+ ε}) \to c_{G} (U)$ as $ε \to 0$ when $U$ is sufficiently regular. The conjunction of (a) and (b) gives $∣ c_{G} (ϕ_{n} (V)) - c_{G} (ϕ (V)) ∣ \to 0$ .

Step 2: $c_{G}$ on symplectic images. Each $ϕ_{n}$ is a symplectomorphism, so $c_{G} (ϕ_{n} (V)) = c_{G} (V)$ by symplectic invariance of the capacity. Passing to the limit using Step 1, $c_{G} (ϕ (V)) = c_{G} (V)$ .

Step 3: Capacity equality forces symplectic. We now claim that the property " $c_{G} (ϕ (V)) = c_{G} (V)$ for every precompact open $V \subset M$ " implies $ϕ^{*} ω = ω$ . Suppose $ϕ^{*} ω \neq = ω$ at some point $p$ . By the local Darboux argument, there exist small ellipsoids $E_{1}, E_{2} \subset M$ centred at $p$ such that $E_{1} ⊊ ϕ^{- 1} (E_{2})$ in $M$ but the Gromov widths satisfy $c_{G} (E_{1}) > c_{G} (ϕ^{- 1} (E_{2}))$ — that is, the non-symplectic distortion of $ϕ$ on a small scale produces an ellipsoid whose capacity is strictly increased under $ϕ^{- 1}$ , violating monotonicity at the limit. (The construction is explicit: one chooses $E_{1}, E_{2}$ as appropriately scaled ellipsoids in the Darboux coordinates aligned with the non-symplectic distortion direction of $ϕ^{*} ω - ω$ , then computes the Gromov width using non-squeezing.)

The conjunction of Step 2 ( $c_{G} (ϕ (V)) = c_{G} (V)$ for every $V$ ) and Step 3 (the contrapositive: capacity equality forces $ϕ$ to be symplectic) closes the argument: $ϕ^{*} ω = ω$ . $□$

Bridge. The proof rests on three ingredients from prior units: the non-squeezing theorem 05.07.01 (which makes the Gromov capacity well-defined and informative), the symplectic capacity 05.07.02 (whose monotonicity and invariance are the abstract input), and pseudoholomorphic curves 05.06.02 (the analytic engine behind non-squeezing). Each ingredient is itself a Gromov-1985 contribution: the announcement of Eliashberg ^{[Eliashberg 1981]} preceded Gromov's published proof by four years, and Gromov's pseudoholomorphic-curve machinery in Pseudoholomorphic curves in symplectic manifolds ^{[Gromov 1985]} supplied both the non-squeezing theorem and the framework in which the capacity-based proof of $C^{0}$ -rigidity becomes natural. Hofer's 1990 alternative proof ^{[Hofer 1990]} runs through the Hofer displacement-energy capacity instead of the Gromov width, illustrating that the rigidity theorem is in no way tied to a single capacity construction — any $C^{0}$ -continuous symplectic invariant that distinguishes balls from cylinders suffices.

Exercises Intermediate+

Exercise 1 (easy, multiple choice).

Let $ϕ_{n} : R^{2} \to R^{2}$ be a sequence of orientation-preserving area-preserving diffeomorphisms converging $C^{0}$ to a diffeomorphism $ϕ$ . Which of the following statements about $ϕ$ is correct?

A. $ϕ$ may fail to be area-preserving, but it is necessarily orientation-preserving
B. $ϕ$ is automatically a symplectomorphism (area-preserving + orientation-preserving)
C. $ϕ$ must be the identity
D. $ϕ$ need not even be a diffeomorphism

Hint

In dimension two, symplectomorphism = area-preserving + orientation-preserving, and Eliashberg-Gromov says this is $C^{0}$ -rigid.

Answer

B. $ϕ$ is automatically a symplectomorphism (area-preserving + orientation-preserving). Eliashberg-Gromov rigidity applies in every dimension, including dimension two. The hypothesis is that $ϕ$ is a diffeomorphism (option D is ruled out by hypothesis); the conclusion is full symplecticness. In dimension two the result was already known to Banyaga-style arguments before Eliashberg-Gromov, but the same conclusion holds with the same proof in arbitrary dimension. Rubric: full credit for B.

Exercise 3 (medium, short-answer).

Show that the linear symplectomorphism $A_{n} (x, y, u, v) = (n x, y / n, u, v)$ of $(R^{4}, ω_{0} = d x \land d y + d u \land d v)$ is a symplectomorphism, then explain why the sequence ${A_{n}}_{n \geq 1}$ does not converge in the $C^{0}$ -topology.

Hint

Verify $A_{n}^{*} ω_{0} = ω_{0}$ on basis vectors, then check uniform convergence on a compact set such as ${∣ x ∣ \leq 1}$ .

Answer

The map $A_{n}$ has matrix $diag (n, 1/ n, 1, 1)$ . Compute $A_{n}^{*} (d x \land d y) = (n d x) \land (d y / n) = d x \land d y$ and $A_{n}^{*} (d u \land d v) = d u \land d v$ , so $A_{n}^{*} ω_{0} = ω_{0}$ and $A_{n}$ is a symplectomorphism. The sequence fails to converge in $C^{0}$ on the compact set $K = {∣ x ∣ \leq 1, ∣ y ∣ \leq 1, ∣ u ∣ \leq 1, ∣ v ∣ \leq 1}$ : $A_{n} (1, 0, 0, 0) = (n, 0, 0, 0)$ has $∣ A_{n} (1, 0, 0, 0) ∣ = n \to \infty$ , so no limit map exists. The example shows that the rigidity theorem applies only when a $C^{0}$ -limit exists, which is itself a substantive constraint on a sequence of symplectomorphisms. Rubric: full credit for verifying the symplectic property and explaining the non-convergence on a single point.

Exercise 4 (medium, short-answer).

State the homeomorphism version of Eliashberg-Gromov rigidity due to Müller-Oh 2007, defining $Sympeo (M, ω)$ along the way, and explain how it differs from the smooth statement.

Hint

The closure of $Symp (M, ω)$ in $Homeo (M)$ under the $C^{0}$ -topology defines a new group of "symplectic homeomorphisms".

Answer

The smooth Eliashberg-Gromov theorem says: $C^{0}$ -limit of symplectomorphisms, if a diffeomorphism, is a symplectomorphism. The Müller-Oh formulation drops the smoothness assumption on the limit: define $Sympeo (M, ω) := \overline{Symp (M, ω)}^{C^{0}} \subset Homeo (M)$ as the $C^{0}$ -closure inside the homeomorphism group. Elements of $Sympeo$ are symplectic homeomorphisms — homeomorphisms that need not be differentiable but admit $C^{0}$ -approximation by symplectomorphisms. Müller-Oh prove $Sympeo$ is a topological group (composition and inversion are $C^{0}$ -continuous), and the smooth Eliashberg-Gromov theorem says $Sympeo (M, ω) \cap Diff (M) = Symp (M, ω)$ . The Müller-Oh formulation is the natural setting for $C^{0}$ -symplectic topology: questions about $C^{0}$ -Hamiltonian flows, Lagrangian rigidity, and quantitative rigidity now take place inside $Sympeo$ . Rubric: full credit for the closure definition, the inclusion-recovery statement, and naming the larger theory.

Exercise 5 (medium, symbolic).

Let $c$ be a symplectic capacity in the sense of 05.07.02, $ϕ \in Diff (M)$ , and suppose $c (ϕ (U)) = c (U)$ for every precompact open $U \subset M$ . Write down the capacity-monotonicity argument that rules out $ϕ^{*} ω \neq = ω$ at any specific point, in one displayed inequality.

Hint

If $ϕ$ contracts in some symplectic direction at a point, a small ellipsoid centred at the point would satisfy $c (ϕ (E)) < c (E)$ .

Answer

The contrapositive: if $ϕ^{*} ω - ω$ is non-zero at $p_{0}$ , choose Darboux coordinates so $ϕ (p_{0}) = 0$ and the non-zero direction of $ϕ^{*} ω - ω$ aligns with $d x_{1} \land d y_{1}$ . The first symplectic plane $Π_{1} = span (\partial_{x_{1}}, \partial_{y_{1}})$ is then contracted under $ϕ$ in symplectic area near $p_{0}$ . Consider an ellipsoid $E (a, R, R, \dots, R) = {∣ z_{1} ∣^{2} / a^{2} + (∣ z_{2} ∣^{2} + \dots + ∣ z_{n} ∣^{2}) / R^{2} \leq 1} \subset M$ where $z_{j} = x_{j} + i y_{j}$ . By Lalonde-McDuff ^{[Hofer-Zehnder]}, for $R$ large and $a$ small, $c_{G} (E (a, R, \dots, R)) = π a^{2}$ . The image $ϕ (E)$ is approximately the ellipsoid with the first axis rescaled by the symplectic contraction factor $λ < 1$ , so

c_{G} (ϕ (E (a, R, \dots, R))) \leq π λ^{2} a^{2} < π a^{2} = c_{G} (E (a, R, \dots, R)),

contradicting the hypothesis $c_{G} (ϕ (U)) = c_{G} (U)$ . Rubric: full credit for the ellipsoid construction and the contradiction inequality.

Exercise 6 (hard, short-answer).

Hofer 1990 On the topological properties of symplectic maps proves rigidity by a different route, using the displacement-energy capacity. Sketch this alternative approach: (a) define the displacement energy of a subset $A$ ; (b) verify the symplectic-invariance and non-degeneracy properties needed to deploy it; (c) explain why the proof structure parallels the Gromov-capacity argument.

Hint

The displacement energy $e (A)$ is the infimum of Hofer norms of Hamiltonian diffeomorphisms that move $A$ off itself.

Answer

(a) Given $A \subset M$ , the Hofer displacement energy is

e (A) := in f {∥ ϕ ∥_{Hofer} : ϕ \in Ham (M, ω), ϕ (A) \cap A = \emptyset},

where the Hofer norm $∥ ϕ ∥_{Hofer} = in f_{H \to ϕ} \int_{0}^{1} (max H_{t} - min H_{t}) d t$ is taken over time-1 Hamiltonian generators. (b) Symplectic invariance: $e (ψ (A)) = e (A)$ for $ψ \in Symp$ . Non-degeneracy: $e (B^{2 n} (r)) > 0$ for any open ball of positive radius (this is Hofer's main analytic input, Proc. R. Soc. Edinburgh 115A ^{[Hofer 1990]}, using pseudoholomorphic-curve estimates). (c) The proof structure parallels Gromov: the displacement energy is $C^{0}$ -continuous on open sets (small $C^{0}$ -perturbations of an isotopy give small Hofer-norm changes), symplectic-invariant, and positive on open balls. The same monotonicity-and-contradiction argument as in the Gromov-capacity proof applies, with $e$ replacing $c_{G}$ . The conceptual point: any $C^{0}$ -continuous symplectic invariant with the right non-degeneracy properties yields a rigidity proof; Gromov 1985 ^{[Gromov 1985]} uses the embedding-based Gromov width; Hofer 1990 uses the dynamical displacement energy; Ekeland-Hofer 1989 Symplectic topology and Hamiltonian dynamics ^{[Ekeland-Hofer 1989]} uses symplectic homology capacities. The rigidity is robust under choice of capacity, reflecting that it is a structural property of the symplectic category rather than an artifact of any one construction. Rubric: full credit for naming the Hofer-norm definition, the displacement-energy capacity, and the parallel proof structure.

Exercise 7 (hard, short-answer).

The Müller-Oh closure $Sympeo (M, ω)$ contains genuinely non-smooth elements ("symplectic homeomorphisms that are not symplectomorphisms"). Sketch how one would construct such an element on $(R^{2 n}, ω_{0})$ , and state what is preserved by $Sympeo$ beyond what is preserved by $Homeo$ .

Hint

Take a sequence of smooth symplectomorphisms approximating a Lipschitz limit that is not $C^{1}$ .

Answer

A standard construction: take a smooth symplectomorphism $ϕ$ on $R^{2 n}$ that is not the identity, and consider the sequence $ϕ_{n} (x) = ϕ (x)$ for $∣ x ∣ \leq n$ extended by the identity smoothly outside (with bump-function interpolation in the annulus $n \leq ∣ x ∣ \leq n + 1$ ). Each $ϕ_{n}$ is a smooth symplectomorphism (the interpolation can be done with $Symp$ -preserving methods, e.g., via the generating-function technology or the Moser trick). The $C^{0}$ -limit is the discontinuous-derivative map $ϕ_{\infty} = ϕ$ on a compact set and identity outside, which is a homeomorphism only after careful gluing — concrete versions of this give a Lipschitz but not $C^{1}$ symplectic homeomorphism. More substantively, Buhovsky-Opshtein 2016 Some quantitative results in $C^{0}$ symplectic geometry ^{[Buhovsky-Opshtein 2016]} construct symplectic homeomorphisms that map smooth Lagrangians to non-smooth (only Lipschitz) embedded submanifolds, exhibiting a strict gap between $Sympeo$ and $Symp$ . The key preserved structures: the Gromov capacity is preserved by $Sympeo$ ( $C^{0}$ -continuity is automatic in the closure); the displacement energy is preserved; symplectic homology and other Floer-theoretic invariants extend to $Sympeo$ via Humilière-Leclercq-Seyfaddini 2015 Reduction of symplectic homeomorphisms ^{[Humilière-Leclercq-Seyfaddini 2015]}. What is not preserved: pointwise differential structure (Jacobians need not exist), the smooth Hamiltonian-flow generators (replaced by $C^{0}$ -Hamiltonian flows in the Müller-Oh sense), and the smooth Lagrangian intersection counts (replaced by $C^{0}$ -Lagrangian intersection numbers, an active research area). Rubric: full credit for an explicit example sketch, listing the capacity-level preserved invariants, and naming the non-smooth pathologies allowed in $Sympeo$ .

Lean formalization Intermediate+

Mathlib has no symplectic-capacity infrastructure and no formalisation of pseudoholomorphic curves, so the Eliashberg-Gromov rigidity theorem is out of formal reach. A proposed signature:

-- Sketch only; no current Mathlib coverage. See lean_mathlib_gap.
import Mathlib.Geometry.Manifold.MFDeriv
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology

variable {M : Type*} [TopologicalSpace M]
  [ChartedSpace (EuclideanSpace ℝ (Fin (2 * n))) M]
  [SmoothManifoldWithCorners _ M]

-- Assuming a SymplecticForm structure with pullback machinery.
structure SymplecticDiffeo (ω : SymplecticForm M) where
  toDiffeo : M ≃ᵈ M
  preserves : toDiffeo.pullback ω = ω

-- The C^0 topology on the diffeomorphism group.
instance : TopologicalSpace (M ≃ᵈ M) := UniformOnFun.topologicalSpace ..

-- Eliashberg-Gromov rigidity, smooth form.
theorem eliashberg_gromov_rigidity (ω : SymplecticForm M)
    (φ : ℕ → SymplecticDiffeo ω) (ψ : M ≃ᵈ M)
    (h : Filter.Tendsto (fun n => (φ n).toDiffeo) Filter.atTop (𝓝 ψ)) :
    ψ.pullback ω = ω := by
  sorry -- depends on Gromov capacities, pseudoholomorphic curves, non-squeezing

Every named structure (SymplecticForm, pullback, SymplecticDiffeo, UniformOnFun.topologicalSpace for $C^{0}$ -convergence on a manifold) is missing or insufficient in Mathlib. The dependency chain reaches back to the pseudoholomorphic-curve infrastructure that underwrites the Gromov-capacity construction itself. Each link in the chain is a multi-year contribution; the integrated rigidity theorem is years beyond formal-library coverage.

Advanced results Master

The Gromov-capacity proof in detail. Gromov's 1985 Pseudoholomorphic curves in symplectic manifolds (Invent. Math. 82) ^{[Gromov 1985]} established the non-squeezing theorem via $J$ -holomorphic disks: if $B^{2 n} (R)$ symplectically embeds into $Z^{2 n} (r)$ , then $R \leq r$ . The Gromov width $c_{G}$ defined by this embedding criterion satisfies $c_{G} (B^{2 n} (r)) = c_{G} (Z^{2 n} (r)) = π r^{2}$ — a substantive equality precisely because the ball and the cylinder have the same Gromov width despite different volumes. Eliashberg's 1981 announcement Rigidity of symplectic and contact structures (Funct. Anal. Appl. 15) ^{[Eliashberg 1981]} preceded Gromov's published proof and outlined the rigidity theorem; Eliashberg's own proof, which used a topological-rigidity argument not yet fully published, was independent. The capacity-based proof presented above (and in McDuff-Salamon Ch. 12) is the consensus modern proof; it routes through the pseudoholomorphic-curve technology Gromov made available.

The Hofer-norm proof (Hofer 1990, Lalonde-McDuff 1995). Helmut Hofer's 1990 On the topological properties of symplectic maps (Proc. R. Soc. Edinburgh 115A) ^{[Hofer 1990]} gave a different proof using the Hofer norm on $Ham (M, ω)$ , the bi-invariant norm

∥ ϕ ∥_{Hofer} := H : [0, 1] \times M \to R, ϕ_{H}^{1} = ϕ in f \int_{0}^{1} (M max H_{t} - M min H_{t}) d t .

The displacement-energy capacity $e (A) = in f {∥ ϕ ∥_{H} : ϕ \in Ham, ϕ (A) \cap A = \emptyset}$ is then $C^{0}$ -continuous and symplectic-invariant; positivity of $e$ on balls is the analytic core, provided by Gromov's pseudoholomorphic-curve estimates and refined by Lalonde-McDuff 1995 The geometry of symplectic energy (Annals of Math. 141). The structure of the rigidity proof is identical to the Gromov-capacity version, illustrating the robustness of the conclusion under choice of $C^{0}$ -continuous symplectic invariant.

Ekeland-Hofer capacities and the capacity hierarchy. Ekeland-Hofer 1989 Symplectic topology and Hamiltonian dynamics (Math. Z. 200) ^{[Ekeland-Hofer 1989]} constructed an infinite sequence $c_{1}^{EH} \leq c_{2}^{EH} \leq \dots$ of symplectic capacities on subdomains of $R^{2 n}$ using variational methods on the symplectic action functional. Each $c_{k}^{EH}$ is monotone, symplectic-invariant, and $C^{0}$ -continuous, so each supplies an independent proof of Eliashberg-Gromov rigidity. The hierarchy

c_{G} \leq c_{HZ} \leq c_{1}^{EH} \leq c_{2}^{EH} \leq \dots

(where $c_{HZ}$ is the Hofer-Zehnder capacity, defined by Hamiltonian-flow periodicity) organises the symplectic invariants and produces refined rigidity statements; the $c_{k}^{EH}$ also detect higher-action periodic orbits and connect to the action spectrum of Hamiltonian diffeomorphisms.

$C^{0}$ -symplectic topology after Müller-Oh. Müller-Oh 2007 The group of Hamiltonian homeomorphisms and $C^{0}$ -symplectic topology (J. Symplectic Geom. 5) ^{[Müller-Oh 2007]} developed the $C^{0}$ -closure programme:

$Sympeo (M, ω) := \overline{Symp (M, ω)}^{C^{0}} \subset Homeo (M)$ , the group of symplectic homeomorphisms.
$Hameo (M, ω) := \overline{Ham (M, ω)}^{(C^{0}, L^{(1, \infty)})}$ , the group of Hamiltonian homeomorphisms, defined as the $C^{0}$ -closure of $Ham$ in $Sympeo$ subject to additional $L^{(1, \infty)}$ -control on the Hofer norm of the approximating sequence (the Müller-Oh "Hofer-Lipschitz" closure).

Both groups are topological groups under their respective topologies. The Müller-Oh definition gives the right setting for $C^{0}$ -Hamiltonian flows: a continuous Hamiltonian $H : [0, 1] \times M \to R$ generates a one-parameter family of Hamiltonian homeomorphisms via a $C^{0}$ -limit of smooth Hamiltonian flows for smooth approximations $H^{ε} \to H$ in $L^{(1, \infty)}$ .

Quantitative $C^{0}$ -rigidity (Buhovsky 2008, Buhovsky-Opshtein 2016). Buhovsky 2008 The 2/3-convergence rate for the Poisson bracket (J. Symplectic Geom. 6) ^{[Buhovsky 2008]} proved that the Poisson bracket ${F, G}$ , viewed as a function on $C^{\infty} (M) \times C^{\infty} (M)$ , is not $C^{0}$ -continuous but is $\frac{2}{3}$ -Hölder continuous in $C^{0}$ at most. The result refines Eliashberg-Polterovich's 2006 $C^{0}$ -rigidity of Poisson brackets (which showed that the Poisson bracket is $C^{0}$ -continuous on $C^{\infty} (M) \times C^{\infty} (M)$ when restricted to bounded $C^{2}$ -norms, a much weaker statement). Buhovsky-Opshtein 2016 Some quantitative results in $C^{0}$ symplectic geometry (Invent. Math. 205) ^{[Buhovsky-Opshtein 2016]} extends to Lagrangian rigidity: the $C^{0}$ -closure of the embedding $L ↪ (M, ω)$ of a smooth Lagrangian is a Lipschitz submanifold that retains its symplectic-area-cohomology class but loses its smooth structure.

Reduction and intersection theory in $Sympeo$ (Humilière-Leclercq-Seyfaddini 2015). Humilière-Leclercq-Seyfaddini 2015 Reduction of symplectic homeomorphisms (Annales scientifiques de l'ENS 48) ^{[Humilière-Leclercq-Seyfaddini 2015]} extends symplectic reduction (the Marsden-Weinstein quotient) to the $C^{0}$ -Hamiltonian setting: a symplectic homeomorphism that preserves the level set of a coisotropic submanifold descends to a symplectic homeomorphism of the reduced space. The result is the foundation for a $C^{0}$ -Floer theory on quotient spaces and for the modern study of $C^{0}$ -Lagrangian intersection theory. Operator-algebraic invariants of $Sympeo$ (Banyaga's flux-style homomorphisms, the Calabi quasi-morphism) are also active research areas.

Connection to mass transport (Buhovsky-Humilière-Seyfaddini 2018). A recent direction connects $C^{0}$ -symplectic topology to optimal transport: Buhovsky-Humilière-Seyfaddini 2018 establish that the Hofer norm is bounded below by an $L^{p}$ -Wasserstein-type distance on the space of probability measures on $M$ , giving a transport-theoretic lower bound on the $C^{0}$ -energy of Hamiltonian flows.

Failure modes.

Capacity-monotonicity arguments require $C^{0}$ -convergence on neighbourhoods, not just at points. The proof of capacity continuity (Step 1 above) breaks down for pointwise-but-not-uniform convergence; the $C^{0}$ hypothesis is essential.
The Gromov width is computed by symplectic-embedding sup, not by infimum or volume. The capacity is the supremal radius of a symplectic ball that fits, not the infimum that contains. Reversing the inequality breaks the monotonicity argument.
Sympeo elements need not be Lipschitz. In the original Müller-Oh definition, symplectic homeomorphisms are allowed to be merely continuous; Lipschitz approximation is a separate hypothesis that gives a strict refinement. The base theorem applies in the unrestricted continuous setting.
The rigidity theorem assumes the limit is a diffeomorphism in its smooth form. The Müller-Oh closure version drops this; without smoothness the conclusion is membership in $Sympeo$ , not in $Symp$ .

Synthesis. Eliashberg-Gromov $C^{0}$ -rigidity is the foundational rigidity statement of the modern symplectic category, isolating the structural fact that the symplectic condition — defined at the level of $C^{1}$ derivatives — admits a $C^{0}$ -continuous characterisation through symplectic capacities. The proof's architecture, articulated by the capacity-based argument of Gromov 1985 ^{[Gromov 1985]} and the displacement-energy argument of Hofer 1990 ^{[Hofer 1990]}, reduces a derivative-level question to a non-derivative-level inequality between $C^{0}$ -continuous functionals. The robustness of the conclusion under choice of capacity (Gromov width, Hofer-Zehnder, Ekeland-Hofer, displacement energy) shows that the rigidity is structural rather than capacity-specific: any $C^{0}$ -continuous symplectic invariant that distinguishes balls from cylinders forces the same conclusion. Within the Müller-Oh extension, the closure $Sympeo (M, ω)$ becomes the natural ambient category for $C^{0}$ -symplectic topology, with the rigidity theorem encoded as the identity $Sympeo (M, ω) \cap Diff (M) = Symp (M, ω)$ . Subsequent developments — Buhovsky's $\frac{2}{3}$ -Hölder bound on the Poisson bracket, Buhovsky-Opshtein's Lagrangian rigidity at the $C^{0}$ -level, Humilière-Leclercq-Seyfaddini's reduction theorem — articulate the $C^{0}$ -category as a substantive object: it has its own dynamical objects ( $C^{0}$ -Hamiltonian flows), its own Floer-theoretic invariants, and its own intersection theory, all preserving structures invisible to the topological-homeomorphism category.

Full proof set Master

The capacity-based proof of Eliashberg-Gromov in dimension $2 n$ , building on the technology developed in 05.07.01 and 05.07.02:

Proposition (capacity continuity under $C^{0}$ -convergence). Let $ϕ_{n} : M \to M$ be a sequence of diffeomorphisms converging $C^{0}$ -uniformly on compact subsets to a diffeomorphism $ϕ$ . For every precompact open set $V \subset M$ , the Gromov widths satisfy $c_{G} (ϕ_{n} (V)) \to c_{G} (ϕ (V))$ .

Proof. Let $ε > 0$ . By $C^{0}$ -convergence on the compact closure $\overline{V}$ , there exists $N$ such that for $n \geq N$ , $d_{g} (ϕ_{n} (x), ϕ (x)) < ε$ for every $x \in \overline{V}$ in the auxiliary Riemannian distance $d_{g}$ . This gives the set-level inclusion $ϕ_{n} (V) \subset (ϕ (V))^{+ ε}$ , the $ε$ -thickening of $ϕ (V)$ , and symmetrically $ϕ (V) \subset (ϕ_{n} (V))^{+ ε}$ . Monotonicity of $c_{G}$ under inclusion gives $c_{G} (ϕ_{n} (V)) \leq c_{G} ((ϕ (V))^{+ ε})$ . As $ε \to 0$ , the thickenings $(ϕ (V))^{+ ε}$ decrease monotonically to $ϕ (V)$ , and upper semicontinuity of $c_{G}$ under monotone open intersections (a standard property of the Gromov width on regular open sets) yields $c_{G} ((ϕ (V))^{+ ε}) \to c_{G} (ϕ (V))$ . The symmetric argument gives the lower bound, completing $c_{G} (ϕ_{n} (V)) \to c_{G} (ϕ (V))$ . $□$

Proposition (symplectic invariance gives limit capacity equality). Under the hypotheses above, if each $ϕ_{n} \in Symp (M, ω)$ , then $c_{G} (ϕ (V)) = c_{G} (V)$ for every precompact open $V \subset M$ .

Proof. For each $n$ , symplectic invariance of $c_{G}$ gives $c_{G} (ϕ_{n} (V)) = c_{G} (V)$ (the right side independent of $n$ ). Passing $n \to \infty$ via the previous proposition, $c_{G} (ϕ (V)) = lim_{n} c_{G} (ϕ_{n} (V)) = c_{G} (V)$ . $□$

Proposition (capacity equality is a symplectic condition at every point). Let $ϕ \in Diff (M)$ satisfy $c_{G} (ϕ (V)) = c_{G} (V)$ for every precompact open $V \subset M$ . Then $\phi^\omega = \omega$.*

Proof. Suppose $ϕ^{*} ω \neq = ω$ , so there is $p_{0} \in M$ with $(ϕ^{*} ω - ω) (p_{0}) \neq = 0$ . In Darboux coordinates centred at $p_{0}$ on a small neighbourhood, $ω$ becomes $ω_{0} = \sum_{i = 1}^{n} d x_{i} \land d y_{i}$ and the linear part $A := D ϕ (p_{0})$ is a linear isomorphism of $T_{p_{0}} M ≅ R^{2 n}$ with $A^{*} ω_{0} \neq = ω_{0}$ .

Consider the symplectic ellipsoid

E (a) := {z \in R^{2 n} : ∣ z_{1} ∣^{2} / a^{2} + ∣ z_{2} ∣^{2} + \dots + ∣ z_{n} ∣^{2} < 1}

for small $a > 0$ centred at $p_{0}$ . By Lalonde-McDuff Hofer's $L^{\infty}$ -geometry: energy and stability of Hamiltonian flows (Annals of Math. 141, 1995) and the original Gromov computation, the Gromov width of $E (a)$ is $c_{G} (E (a)) = π a^{2}$ for $a$ small enough that the $z_{1}$ -axis is the constraining direction. The image $A (E (a))$ is the linear-image ellipsoid; if $A^{*} ω_{0} \neq = ω_{0}$ , there is a symplectic direction in $R^{2 n}$ along which $A$ either contracts or dilates the symplectic area form. In the contracting case, $A (E (a))$ sits inside a smaller ellipsoid whose Gromov width is strictly less than $π a^{2}$ ; in the dilating case, the inverse $A^{- 1}$ contracts, and the same argument applies to $ϕ^{- 1}$ .

For $V = E (a)$ centred at $p_{0}$ with $a$ chosen small enough that $ϕ$ is well-approximated by its linear part $A$ on $V$ , the resulting capacity computation gives $c_{G} (ϕ (V)) < c_{G} (V)$ , contradicting the hypothesis. Hence $ϕ^{*} ω = ω$ on a dense set of points, and by continuity of $ϕ^{*} ω$ on the whole of $M$ . $□$

Theorem (Eliashberg-Gromov, smooth form). The subgroup $Symp (M, ω) \subset Diff (M)$ is $C^{0}$ -closed in $Diff (M)$ .

Proof. Let $ϕ_{n} \in Symp (M, ω)$ converge $C^{0}$ -uniformly on compact subsets to $ϕ \in Diff (M)$ . By the first two propositions, $c_{G} (ϕ (V)) = c_{G} (V)$ for every precompact open $V$ . By the third proposition, $ϕ^{*} ω = ω$ , so $ϕ \in Symp (M, ω)$ . $□$

Stated without proof — see primary citation.

The non-squeezing theorem $c_{G} (B^{2 n} (r)) = c_{G} (Z^{2 n} (r)) = π r^{2}$ : Gromov 1985 ^{[Gromov 1985]}, via $J$ -holomorphic disks. Proved in 05.07.01.
The well-definedness and basic properties of the Gromov width as a symplectic capacity: Gromov 1985 and McDuff-Salamon Ch. 12. Proved in 05.07.02.
The positivity of the Hofer displacement energy on open balls: Hofer 1990 ^{[Hofer 1990]} and Lalonde-McDuff 1995 The geometry of symplectic energy (Annals of Math. 141, 349-371).
The Ekeland-Hofer capacities $c_{k}^{EH}$ and their independence properties: Ekeland-Hofer 1989 ^{[Ekeland-Hofer 1989]}.
The Müller-Oh closure $Sympeo$ as a topological group: Müller-Oh 2007 ^{[Müller-Oh 2007]}.
The $\frac{2}{3}$ -Hölder bound on the Poisson bracket: Buhovsky 2008 ^{[Buhovsky 2008]}.
The reduction theorem for symplectic homeomorphisms: Humilière-Leclercq-Seyfaddini 2015 ^{[Humilière-Leclercq-Seyfaddini 2015]}.

Connections Master

Gromov non-squeezing theorem 05.07.01. The non-squeezing theorem produces the Gromov width as a symplectic capacity that distinguishes balls from cylinders; without non-squeezing, $c_{G}$ would coincide with the volume capacity and the rigidity theorem would degenerate. The rigidity theorem is the "global" consequence of the "local" non-squeezing constraint: non-squeezing says that capacities exist and discriminate, rigidity says that capacities are $C^{0}$ -continuous enough to force the symplectic condition on $C^{0}$ -limits.
Symplectic capacity 05.07.02. The Gromov width is the prototypical symplectic capacity, and the rigidity proof uses precisely the three axioms of a capacity — monotonicity, symplectic invariance, and the ball-cylinder normalisation. The full hierarchy of capacities $c_{G} \leq c_{HZ} \leq c_{1}^{EH} \leq c_{2}^{EH} \leq \dots$ each independently yields a rigidity proof, illustrating the structural nature of the result.
Pseudoholomorphic curve 05.06.02. The non-squeezing theorem (and hence the Gromov-width existence) rests on $J$ -holomorphic curves in symplectic manifolds. Gromov's 1985 introduction of pseudoholomorphic curves made both non-squeezing and rigidity accessible; the analytic compactness and bubbling theorems for $J$ -holomorphic curves are the technical foundation of the Gromov capacity construction.
Floer homology 05.08.02. Floer-theoretic invariants of symplectic manifolds — symplectic homology, Lagrangian Floer homology, Hamiltonian Floer homology — extend to the $C^{0}$ -symplectic category via the Müller-Oh closure and the Humilière-Leclercq-Seyfaddini reduction theorem. The rigidity theorem is the prerequisite that makes the $C^{0}$ -extension well-defined as a category.
Arnold conjecture 05.08.01. The Arnold conjecture in its $C^{0}$ -Hamiltonian form (the closure version) refines the smooth Hamiltonian conjecture: $C^{0}$ -Hamiltonian diffeomorphisms in the Müller-Oh sense satisfy analogous lower bounds for fixed-point counts. The rigidity theorem is what allows this extension to be formulated; without it, the closure $Hameo$ would not have well-defined Hofer norms or action spectra.
Contact topology and Reeb dynamics 05.10.04. The Eliashberg-Gromov rigidity philosophy has a contact analogue: the contactomorphism group is $C^{0}$ -closed in the diffeomorphism group, with similar capacity-style arguments. The contact rigidity statement is the foundation of $C^{0}$ -contact topology, an active research area.
Symplectic manifold 05.01.02. The rigidity theorem applies to every symplectic manifold $(M, ω)$ ; it is one of the foundational properties that makes the symplectic category distinct from the volume-form category or the topological-manifold category. The closure $Sympeo (M, ω)$ is the natural setting for studying $(M, ω)$ up to $C^{0}$ -deformation.

Historical & philosophical context Master

Yakov Eliashberg announced the $C^{0}$ -rigidity theorem in 1981 Rigidity of symplectic and contact structures (Funct. Anal. Appl. 15) ^{[Eliashberg 1981]} with a topological argument that did not appear in published form for many years. Eliashberg's original approach used a careful study of the topology of the symplectomorphism group via path-spaces, anticipating but not fully reaching the capacity-theoretic framework. Mikhail Gromov's 1985 Pseudoholomorphic curves in symplectic manifolds (Inventiones Math. 82) ^{[Gromov 1985]} independently established the non-squeezing theorem and outlined the capacity-based proof of rigidity that has become standard. Gromov's introduction of pseudoholomorphic curves into symplectic geometry transformed the field, making rigidity statements like Eliashberg-Gromov accessible via analytic methods that previously did not exist in the subject.

Helmut Hofer's 1990 On the topological properties of symplectic maps (Proc. R. Soc. Edinburgh 115A) ^{[Hofer 1990]} gave the alternative rigidity proof using the displacement-energy capacity and the bi-invariant Hofer norm on the Hamiltonian group; the Hofer norm is named after this paper, and the norm's existence (as a non-degenerate bi-invariant metric) is itself an important rigidity statement about $Ham (M, ω)$ . Ivar Ekeland and Helmut Hofer's 1989 Symplectic topology and Hamiltonian dynamics (Math. Z. 200) ^{[Ekeland-Hofer 1989]} constructed the Ekeland-Hofer capacity hierarchy via variational analysis on the symplectic action functional, providing an infinite family of $C^{0}$ -continuous symplectic invariants and embedding the rigidity theorem in a richer hierarchy of refinements.

The $C^{0}$ -symplectic-topology programme began with Yong-Geun Oh and Stefan Müller's 2007 The group of Hamiltonian homeomorphisms and $C^{0}$ -symplectic topology (J. Symplectic Geometry 5) ^{[Müller-Oh 2007]}, which defined the closures $Sympeo (M, ω)$ and $Hameo (M, ω)$ as topological groups and recast the rigidity theorem as a structural identity for the new closures. Lev Buhovsky's 2008 The 2/3-convergence rate for the Poisson bracket (J. Symplectic Geometry 6) ^{[Buhovsky 2008]} supplied the quantitative refinement of Eliashberg-Polterovich 2006 $C^{0}$ -continuity of the Poisson bracket, showing that the Poisson bracket admits at best a $\frac{2}{3}$ -Hölder modulus of continuity in $C^{0}$ . Lev Buhovsky and Emmanuel Opshtein's 2016 Some quantitative results in $C^{0}$ symplectic geometry (Inventiones Math. 205) ^{[Buhovsky-Opshtein 2016]} established Lagrangian $C^{0}$ -rigidity in a quantitative form and constructed explicit examples of non-smooth symplectic homeomorphisms.

Vincent Humilière, Rémi Leclercq, and Sobhan Seyfaddini's 2015 Reduction of symplectic homeomorphisms (Annales scientifiques de l'ENS 48) ^{[Humilière-Leclercq-Seyfaddini 2015]} extended symplectic reduction to the Müller-Oh setting and supplied the foundational structural results that make $C^{0}$ -Floer theory tractable. The modern programme studies $C^{0}$ -Lagrangian intersection numbers, $C^{0}$ -Hamiltonian dynamics, and the Calabi quasi-morphism on $Hameo$ with techniques inherited from Floer theory but applied at the level of the $C^{0}$ -closure.

The Eliashberg-Gromov theorem is now a cornerstone of symplectic topology: it sits between the local Darboux normal-form theorem (which says that symplectic structure has no local invariants beyond dimension) and the deep global rigidity statements of Gromov-Floer-Hofer (which say that the symplectic category has subtle global invariants invisible to topology alone). The rigidity asserts that the space of symplectic structures, viewed inside the space of all $C^{0}$ -data, is itself rigid — a foundational property that distinguishes symplectic geometry from the much more flexible volume-preserving category.

Bibliography Master

@article{Eliashberg1981Rigidity,
  author  = {Eliashberg, Yakov},
  title   = {Rigidity of symplectic and contact structures (announcement)},
  journal = {Funct. Anal. Appl.},
  volume  = {15},
  year    = {1981},
  pages   = {145--153}
}

@article{Gromov1985PseudoHolomorphic,
  author  = {Gromov, Mikhail},
  title   = {Pseudo holomorphic curves in symplectic manifolds},
  journal = {Invent. Math.},
  volume  = {82},
  year    = {1985},
  pages   = {307--347}
}

@article{Hofer1990Rigidity,
  author  = {Hofer, Helmut},
  title   = {On the topological properties of symplectic maps},
  journal = {Proc. R. Soc. Edinburgh Sect. A},
  volume  = {115},
  year    = {1990},
  pages   = {25--38}
}

@article{EkelandHofer1989,
  author  = {Ekeland, Ivar and Hofer, Helmut},
  title   = {Symplectic topology and {H}amiltonian dynamics},
  journal = {Math. Z.},
  volume  = {200},
  year    = {1989},
  pages   = {355--378}
}

@book{HoferZehnder1994,
  author    = {Hofer, Helmut and Zehnder, Eduard},
  title     = {Symplectic Invariants and {H}amiltonian Dynamics},
  series    = {Birkh{\"a}user Advanced Texts},
  publisher = {Birkh{\"a}user Verlag},
  year      = {1994}
}

@book{McDuffSalamon2017,
  author    = {McDuff, Dusa and Salamon, Dietmar},
  title     = {Introduction to Symplectic Topology},
  edition   = {3},
  publisher = {Oxford University Press},
  year      = {2017}
}

@article{MullerOh2007,
  author  = {M{\"u}ller, Stefan and Oh, Yong-Geun},
  title   = {The group of {H}amiltonian homeomorphisms and {$C^0$}-symplectic topology},
  journal = {J. Symplectic Geom.},
  volume  = {5},
  year    = {2007},
  pages   = {167--219}
}

@article{Buhovsky2008Poisson,
  author  = {Buhovsky, Lev},
  title   = {The 2/3-convergence rate for the {P}oisson bracket},
  journal = {J. Symplectic Geom.},
  volume  = {6},
  year    = {2008},
  pages   = {139--162}
}

@article{BuhovskyOpshtein2016,
  author  = {Buhovsky, Lev and Opshtein, Emmanuel},
  title   = {Some quantitative results in {$C^0$} symplectic geometry},
  journal = {Invent. Math.},
  volume  = {205},
  year    = {2016},
  pages   = {1--56}
}

@article{HumiliereLeclercqSeyfaddini2015,
  author  = {Humili{\`e}re, Vincent and Leclercq, R{\'e}mi and Seyfaddini, Sobhan},
  title   = {Reduction of symplectic homeomorphisms},
  journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r.},
  volume  = {48},
  year    = {2015},
  pages   = {633--668}
}

@article{LalondeMcDuff1995,
  author  = {Lalonde, Fran{\c{c}}ois and McDuff, Dusa},
  title   = {The geometry of symplectic energy},
  journal = {Ann. of Math. (2)},
  volume  = {141},
  year    = {1995},
  pages   = {349--371}
}

@article{EliashbergPolterovich2006,
  author  = {Eliashberg, Yakov and Polterovich, Leonid},
  title   = {{$C^0$}-rigidity of {P}oisson brackets},
  journal = {Algebr. Geom. Topol.},
  volume  = {6},
  year    = {2006},
  pages   = {2491--2509}
}

Prerequisites

05.07.01
05.07.02
05.06.02

Tier anchors

beginner: McDuff-Salamon §12.2 informal; Hofer-Zehnder §5 introductory pages
intermediate: McDuff-Salamon Ch. 12; Hofer-Zehnder *Symplectic Invariants and Hamiltonian Dynamics* §5
master: Eliashberg 1981 *Funct. Anal. Appl.* 15 (announcement); Gromov 1985 *Invent. Math.* 82; Hofer 1990 *Proc. R. Soc. Edinburgh* 115A; Ekeland-Hofer 1989 *Math. Z.* 200; Müller-Oh 2007 *J. Symplectic Geom.* 5; Buhovsky 2008 *J. Symplectic Geom.* 6; Humilière-Leclercq-Seyfaddini 2015; Buhovsky-Opshtein 2016

References

Eliashberg 1981 — Rigidity of symplectic and contact structures (announcement) · Funct. Anal. Appl. 15, 145-153 (originating $C^0$-rigidity announcement)
Gromov 1985 — Pseudoholomorphic curves in symplectic manifolds · Invent. Math. 82, 307-347 (independent capacity-based proof; non-squeezing)
Hofer 1990 — On the topological properties of symplectic maps · Proc. R. Soc. Edinburgh 115A, 25-38 (Hofer-norm proof; displacement-energy capacity)
Ekeland-Hofer 1989 — Symplectic topology and Hamiltonian dynamics · Math. Z. 200, 355-378 (capacity hierarchy; alternative rigidity proof)
Hofer-Zehnder — Symplectic Invariants and Hamiltonian Dynamics · Birkhäuser 1994, §5 (textbook presentation of $C^0$-rigidity via capacities)
McDuff-Salamon — Introduction to Symplectic Topology · Oxford 3rd ed. 2017, Ch. 12 (rigidity via Hofer-Zehnder capacity)
Müller-Oh 2007 — The group of Hamiltonian homeomorphisms and $C^0$-symplectic topology · J. Symplectic Geom. 5, 167-219 (definition of $\mathrm{Sympeo}$ and $\mathrm{Hameo}$)
Buhovsky 2008 — The 2/3-convergence rate for the Poisson bracket · J. Symplectic Geom. 6, 139-162 (refined $C^0$-rigidity of Poisson brackets)
Humilière-Leclercq-Seyfaddini 2015 — Reduction of symplectic homeomorphisms · Annales scientifiques de l'ENS 48, 633-668 (modern $C^0$-symplectic topology)
Buhovsky-Opshtein 2016 — Some quantitative results in $C^0$ symplectic geometry · Invent. Math. 205, 1-56 (Lagrangian $C^0$-rigidity, quantitative refinements)

Estimated time

beginner: 16m
intermediate: 40m
master: 70m