🔵Symplectic Geometry Unit 10 – Gromov's Non-Squeezing & Symplectic Capacities

What's the Big Idea?

• Gromov's non-squeezing theorem is a fundamental result in symplectic geometry that highlights the rigidity of symplectic mappings
• The theorem states that a symplectic embedding of a ball into a cylinder is impossible if the radius of the ball is greater than the radius of the cylinder
• This result demonstrates that symplectic mappings preserve certain geometric properties, unlike volume-preserving mappings
• Symplectic capacities are numerical invariants that quantify the size of symplectic manifolds and help distinguish between them
• The existence of symplectic capacities is a consequence of Gromov's non-squeezing theorem
  • They provide a way to measure the "symplectic size" of a manifold
  • Different capacities can give different information about the symplectic structure
• Gromov's work laid the foundation for the development of symplectic topology as a distinct field of study

Key Concepts to Grasp

• Symplectic manifolds are even-dimensional manifolds equipped with a closed, non-degenerate 2-form called the symplectic form
  • The symplectic form $\omega$ satisfies $d\omega = 0$ (closed) and $\omega^n \neq 0$ (non-degenerate), where $2n$ is the dimension of the manifold
• Symplectic mappings are transformations that preserve the symplectic form
  • They are diffeomorphisms $\phi: (M, \omega) \to (N, \omega')$ such that $\phi^* \omega' = \omega$
• Symplectic embeddings are symplectic mappings that are injective (one-to-one)
• The standard symplectic vector space is $\mathbb{R}^{2n}$ with the symplectic form $\omega_0 = \sum_{i=1}^n dx_i \wedge dy_i$
• A symplectic capacity is a function $c$ that assigns a non-negative number to each symplectic manifold and satisfies certain properties
  • Monotonicity: If there exists a symplectic embedding $\phi: (M, \omega) \to (N, \omega')$, then $c(M, \omega) \leq c(N, \omega')$
  • Conformality: For any symplectic manifold $(M, \omega)$ and $\lambda > 0$, $c(M, \lambda\omega) = \lambda c(M, \omega)$
  • Normalization: For the standard ball $B^{2n}(r)$ and cylinder $Z^{2n}(r) = B^2(r) \times \mathbb{R}^{2n-2}$, $c(B^{2n}(1)) = c(Z^{2n}(1)) = \pi$

The Math Behind It

• Gromov's non-squeezing theorem can be stated mathematically as follows:
  • If there exists a symplectic embedding $\phi: (B^{2n}(r), \omega_0) \to (Z^{2n}(R), \omega_0)$, then $r \leq R$
• The proof of the non-squeezing theorem relies on the notion of pseudoholomorphic curves and the compactness properties of the moduli space of such curves
• Pseudoholomorphic curves are maps $u: (\Sigma, j) \to (M, J)$ from a Riemann surface $(\Sigma, j)$ to an almost complex manifold $(M, J)$ that satisfy the Cauchy-Riemann equation $du \circ j = J \circ du$
  • In the context of symplectic geometry, the almost complex structure $J$ is compatible with the symplectic form $\omega$
• The moduli space of pseudoholomorphic curves has a natural compactification, the Gromov compactification, which includes nodal curves
• The existence of a symplectic embedding $\phi: (B^{2n}(r), \omega_0) \to (Z^{2n}(R), \omega_0)$ with $r > R$ would imply the existence of a pseudoholomorphic curve in $Z^{2n}(R)$ with certain properties, leading to a contradiction

Real-World Applications

• Symplectic geometry has found applications in various branches of physics, including classical mechanics, quantum mechanics, and string theory
• In classical mechanics, the phase space of a system is a symplectic manifold, and the evolution of the system is governed by symplectic mappings (Hamiltonian flows)
  • Gromov's non-squeezing theorem implies that certain phase space transformations are impossible, providing constraints on the dynamics of the system
• In quantum mechanics, the state space of a quantum system is a projective Hilbert space, which has a natural symplectic structure (the Fubini-Study form)
  • Symplectic techniques have been used to study the geometry of quantum entanglement and the properties of quantum channels
• In string theory, the target space of a sigma model is often a symplectic manifold, and the study of pseudoholomorphic curves plays a crucial role in understanding the properties of the theory
  • Gromov-Witten invariants, which count pseudoholomorphic curves, have been used to compute certain physical quantities in string theory, such as the partition function

Mind-Bending Examples

• The Gromov-Eliashberg theorem states that the group of symplectomorphisms (symplectic diffeomorphisms) of a symplectic manifold is C0-closed in the group of all diffeomorphisms
  • This means that a sequence of symplectomorphisms converging uniformly to a diffeomorphism must converge to a symplectomorphism
  • In contrast, the group of volume-preserving diffeomorphisms is not C0-closed, demonstrating the rigidity of symplectic mappings
• The symplectic camel theorem, a corollary of Gromov's non-squeezing theorem, states that a symplectic ball cannot be squeezed through a hole in a symplectic cylinder if the radius of the ball is greater than the radius of the hole
  • This is reminiscent of the saying "it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God" (Matthew 19:24)
• The Hofer-Zehnder capacity, a specific symplectic capacity, can be used to detect the existence of periodic orbits in Hamiltonian systems
  • For example, if a symplectic manifold has a compact subset with a small Hofer-Zehnder capacity, then any Hamiltonian function attaining its maximum on that subset must have a periodic orbit

Common Pitfalls and How to Avoid Them

• Confusing symplectic mappings with volume-preserving mappings
  • While symplectic mappings preserve volume (Liouville's theorem), not all volume-preserving mappings are symplectic
  • To avoid this confusion, always check that the mapping preserves the symplectic form, not just the volume element
• Forgetting the non-degeneracy condition when working with symplectic forms
  • A closed 2-form $\omega$ is symplectic only if $\omega^n \neq 0$ everywhere, where $2n$ is the dimension of the manifold
  • Always verify that the non-degeneracy condition holds when dealing with symplectic structures
• Misunderstanding the role of almost complex structures in symplectic geometry
  • Not every symplectic manifold admits a compatible almost complex structure (e.g., the symplectic torus $\mathbb{T}^{2n}$ for $n > 1$)
  • When working with pseudoholomorphic curves, ensure that the almost complex structure is compatible with the symplectic form
• Overlooking the importance of the compactness properties of the moduli space of pseudoholomorphic curves
  • The Gromov compactification is crucial for the proof of the non-squeezing theorem and other results in symplectic topology
  • When studying moduli spaces of pseudoholomorphic curves, always consider their compactification and the possible presence of nodal curves

Connecting the Dots

• Gromov's non-squeezing theorem and symplectic capacities are intimately related to other important concepts in symplectic geometry, such as Hofer geometry and displacement energy
• Hofer geometry is a metric geometry on the group of Hamiltonian diffeomorphisms of a symplectic manifold, where the distance between two diffeomorphisms is measured by the minimal Hofer norm of a Hamiltonian function generating a path between them
  • The Hofer norm of a Hamiltonian function is defined using the oscillation of the function, which is related to the displacement energy of subsets of the symplectic manifold
• The displacement energy of a subset $A$ of a symplectic manifold $(M, \omega)$ is the infimum of the Hofer norms of Hamiltonian functions $H$ such that $\phi_H^1(A) \cap A = \emptyset$, where $\phi_H^1$ is the time-1 flow of the Hamiltonian vector field of $H$
  • Symplectic capacities provide lower bounds for the displacement energy of subsets, and the non-squeezing theorem can be interpreted as a statement about the displacement energy of symplectic balls
• The study of symplectic capacities and displacement energy has led to the development of symplectic rigidity theory, which investigates the extent to which symplectic mappings are constrained by the geometry of the underlying symplectic manifold
  • Symplectic rigidity results, such as the non-squeezing theorem and the Hofer-Zehnder capacity, demonstrate that symplectic mappings are much more rigid than volume-preserving mappings, and this rigidity is a fundamental feature of symplectic geometry

Going Beyond the Basics

• The Gromov-Witten invariants, which count pseudoholomorphic curves in symplectic manifolds, provide a powerful tool for studying the topology of symplectic manifolds and their Lagrangian submanifolds
  • These invariants have been used to define symplectic capacities, such as the Gromov width, and to prove existence results for symplectic embeddings and Lagrangian intersections
  • The theory of Gromov-Witten invariants has also been extended to the realm of Floer homology, which is a powerful tool for studying the dynamics of Hamiltonian systems and the geometry of Lagrangian submanifolds
• The Fukaya category of a symplectic manifold is an A∞-category whose objects are Lagrangian submanifolds and whose morphisms are Floer homology groups
  • The Fukaya category encodes the symplectic topology of the manifold and has been used to study mirror symmetry, a conjectural duality between symplectic and complex geometry
  • The study of Fukaya categories and their relationship to Gromov-Witten theory has led to the development of homological mirror symmetry, which has become a major research area in modern mathematics
• The Eliashberg-Gromov theorem on the C0-closedness of the group of symplectomorphisms has been generalized to the contact setting, where the group of contactomorphisms is also C0-closed
  • This result has important applications in the study of Reeb dynamics and the geometry of contact manifolds
  • The proof of the Eliashberg-Gromov theorem in the contact setting relies on the notion of symplectic field theory (SFT), a powerful framework that combines ideas from symplectic topology, contact geometry, and string topology
• The study of symplectic capacities and displacement energy has also been extended to the setting of Hamiltonian group actions, where the notion of equivariant symplectic capacities and equivariant displacement energy has been developed
  • These equivariant invariants have been used to study the geometry of symplectic quotients and the existence of periodic orbits in symmetric Hamiltonian systems
  • The theory of equivariant symplectic capacities has also been applied to the study of symplectic packing problems and the geometry of symplectic toric manifolds