10.1 Statement and proof of Gromov's non-squeezing theorem
3 min read•july 30, 2024
is a cornerstone of symplectic geometry. It shows you can't squeeze a big ball into a skinny cylinder symplectically, even if the volumes match up. This reveals a fundamental rigidity in symplectic spaces that goes beyond .
The proof uses , a powerful technique in symplectic geometry. By cleverly choosing an and analyzing curve areas, we can show the impossibility of certain symplectic embeddings, highlighting the unique properties of symplectic transformations.
Geometric intuition for Gromov's theorem
Fundamental rigidity in symplectic geometry
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- Gromov's non-squeezing theorem demonstrates impossibility of symplectically embedding large ball into thin cylinder even when ball's volume smaller than cylinder's
- Contrasts with flexibility in volume-preserving transformations highlights fundamental rigidity in symplectic geometry
- Implies symplectic transformations preserve geometric structures beyond volume
- Visualized as inability to squeeze fat object through narrow opening while maintaining symplectic structure
- Relates to symplectic capacities measuring symplectic size of sets differently from volume
- Necessitates preservation of certain two-dimensional projections by symplectic embeddings not just overall 2n-dimensional volume
Conceptual implications
- Reveals deeper geometric constraints in symplectic spaces ( in classical mechanics)
- Suggests existence of invariants in symplectic geometry beyond volume (symplectic capacities)
- Indicates symplectic transformations respect certain area-preserving properties in phase space projections
- Demonstrates fundamental difference between symplectic and volume-preserving maps
- Implies limitations on possible deformations of phase space regions under
- Provides insight into geometric nature of symplectic manifolds and their distinct properties
Proof of Gromov's non-squeezing theorem
Pseudo-holomorphic curves technique
- Utilizes J-holomorphic curves in almost complex manifolds
- Involves choosing almost complex structure J on R^2n
- Compatible with standard
- Agrees with standard complex structure outside compact set
- Considers of J-holomorphic curves with boundary on ball and cylinder
- Relies on to show moduli space compactness
- Demonstrates existence of J-holomorphic curve passing through ball center and reaching cylinder boundary
Contradiction and area arguments
- Uses area of J-holomorphic curve to derive contradiction if ball radius exceeds cylinder radius
- Relates symplectic area of J-holomorphic curve to radii of ball and cylinder
- Establishes non-squeezing result through area comparison
- Employs properties of symplectic form and J-holomorphic curves to bound areas
- Leverages relationship between symplectic and complex structures in proof
Significance of Gromov's theorem
Foundational impact on symplectic geometry
- Represents fundamental result highlighting rigidity of symplectic structures
- Provides clear distinction between volume-preserving and symplectic transformations
- Led to development of symplectic capacities (important invariants)
- Serves as cornerstone for numerous results in (symplectic embeddings, packing problems)
- Opened new research avenues in symplectic geometry through J-holomorphic curves technique
- Crucial for understanding nature of symplectic manifolds and their unique geometric properties
Applications and implications
- Constrains possible evolutions of phase space under Hamiltonian flows
- Impacts understanding of Hamiltonian dynamics in classical mechanics
- Provides insights into quantitative aspects of symplectic geometry
- Influences study of symplectic invariants and their properties
- Affects analysis of symplectic capacities and their relationships
- Shapes understanding of geometric constraints in phase spaces of physical systems
Applications of Gromov's theorem
Symplectic embeddings and packing problems
- Provides necessary condition for existence of symplectic embeddings between certain domains
- Proves non-existence of symplectic embeddings where volume considerations suggest possibility
- Determines lower bounds for symplectic capacities of various symplectic manifolds and subsets
- Solves problems related to symplectic packing (maximum number of equal-sized balls symplectically embedded into given manifold)
- Offers quantitative bounds on symplectic embeddings and deformations
- Aids in analyzing symplectic geometry of phase spaces in classical mechanics (billiard systems)
Hamiltonian systems and phase space analysis
- Constrains possible evolutions of phase space regions in Hamiltonian systems
- Provides tool for understanding symplectic geometry of phase spaces in physics (plasma physics)
- Impacts analysis of Hamiltonian flows and their long-term behavior
- Influences study of stability and instability in Hamiltonian systems
- Aids in understanding geometric aspects of integrable and non-integrable systems
- Contributes to analysis of symplectic capacities in physical systems (quantum mechanics)
Key Terms to Review (25)
Action-angle coordinates: Action-angle coordinates are a powerful tool in symplectic geometry, used to describe integrable systems where the dynamics can be expressed in terms of action variables and angle variables. These coordinates transform the Hamiltonian system into a simpler form, revealing the underlying structure of the system, especially in the context of conservation laws and periodic motion. They are particularly significant in understanding how symplectic manifolds relate to integrable systems and serve as a foundation for various important results in geometry and physics.
Almost Complex Structure: An almost complex structure on a smooth manifold is a (1,1)-tensor field that satisfies a specific algebraic condition, making it compatible with the manifold's differentiable structure. This allows for the definition of complex-like geometry on real manifolds, which can be crucial for understanding their symplectic properties and relationships with complex manifolds. Such structures play a vital role in various geometric theories, including the formulation of Gromov's non-squeezing theorem.
Capacity: In the context of symplectic geometry, capacity is a measure of the size or 'volume' of a symplectic manifold, capturing the idea of how much a certain geometric shape can hold. It serves as a critical concept when discussing embeddings and transformations in the setting of Gromov's non-squeezing theorem, which states that certain shapes cannot be squeezed into smaller forms without violating their capacity.
Contact Geometry: Contact geometry is a branch of differential geometry that studies contact structures on odd-dimensional manifolds. These structures can be thought of as a way to define a 'hyperplane' at each point of the manifold, providing a geometric framework that captures the behavior of dynamical systems and their trajectories. Contact geometry plays a significant role in understanding symplectomorphisms, symplectic capacities, and is closely related to Gromov's non-squeezing theorem, as it provides insights into the geometric properties of embeddings and constraints in symplectic manifolds.
Continuous Deformation: Continuous deformation refers to a process where a geometric object is transformed into another shape without any sudden jumps or breaks. This concept is essential in topology and symplectic geometry, as it allows for the study of properties that remain invariant under such smooth transformations, including the context of Gromov's non-squeezing theorem, which deals with the ways shapes can be manipulated while preserving certain characteristics.
Cylindrical Symplectic Manifold: A cylindrical symplectic manifold is a specific type of symplectic manifold that has a cylindrical structure, typically expressed as a symplectic form that can be decomposed into two parts: one that varies along the cylindrical direction and another that is constant. This structure allows for a deeper understanding of the behavior of symplectic embeddings and plays a critical role in Gromov's non-squeezing theorem, which addresses how symplectic volumes interact with geometric shapes in these manifolds.
Darbu's Theorem: Darbu's Theorem states that if a symplectic manifold is equipped with a Hamiltonian function, then the orbits of the corresponding Hamiltonian flow can be understood in terms of the properties of the underlying symplectic structure. This theorem connects the Hamiltonian formalism with Lagrangian mechanics by demonstrating how solutions to Hamilton's equations reflect geometric properties of symplectic manifolds. It also plays a role in understanding the behavior of dynamical systems in relation to Gromov's non-squeezing theorem.
Embedding problems: Embedding problems refer to questions regarding the possibility of smoothly and symplectically embedding one geometric or topological space into another. These issues are particularly significant in symplectic geometry as they explore the limitations and capabilities of certain spaces to contain or represent others while preserving their essential geometric structures.
Gromov Compactness Theorem: The Gromov Compactness Theorem is a result in differential geometry that states a family of Riemannian manifolds can be compact if their metrics are uniformly bounded and the curvature is uniformly bounded below. This theorem plays a crucial role in the study of symplectic geometry and geometric analysis, particularly in understanding the convergence properties of sequences of manifolds.
Gromov's Non-Squeezing Theorem: Gromov's Non-Squeezing Theorem states that a symplectic manifold cannot be 'squeezed' into a smaller symplectic volume than it originally has, specifically, a ball in a symplectic space cannot be symplectically embedded into a narrower cylinder unless the cylinder has at least the same volume. This theorem highlights fundamental limitations on how symplectic structures can be manipulated, connecting various concepts in symplectic geometry and its applications in both mathematics and physics.
Hamiltonian Flows: Hamiltonian flows describe the evolution of a dynamical system defined by Hamiltonian mechanics, where the system's evolution is governed by a Hamiltonian function. This function represents the total energy of the system, and the flows provide a way to analyze how physical systems evolve over time in a symplectic space. Understanding Hamiltonian flows is essential for grasping the behavior of systems in terms of energy conservation and phase space dynamics.
J-holomorphic curves: j-holomorphic curves are smooth mappings from a Riemann surface into a symplectic manifold, which are holomorphic with respect to a compatible almost complex structure j. These curves play a central role in symplectic geometry and have significant implications in areas such as Gromov's non-squeezing theorem and the study of pseudo-holomorphic invariants in symplectic topology.
Liouville Domain: A Liouville domain is a type of symplectic manifold that has a specific geometric structure characterized by a one-form that is exact and has a non-negative integral over any simple closed curve. This property implies that it admits a certain type of 'filling' by symplectic manifolds, playing a crucial role in symplectic topology and the formulation of Gromov's non-squeezing theorem.
Mikhail Gromov: Mikhail Gromov is a prominent Russian mathematician known for his significant contributions to various areas of mathematics, including symplectic geometry, topology, and geometric group theory. He is particularly famous for developing concepts such as Gromov's non-squeezing theorem, which has implications in the study of symplectic manifolds and the relationships between geometric structures.
Moduli space: A moduli space is a geometric space that parametrizes a certain class of mathematical objects, allowing for the classification of these objects based on specific properties. In symplectic geometry, moduli spaces often arise in the study of geometric structures and invariants, as they help to understand the relationships between different configurations, such as the non-squeezing phenomenon described by Gromov's theorem.
Moser's Theorem: Moser's Theorem states that if a symplectic manifold possesses a symplectic structure that is compatible with a Hamiltonian function, then any two Hamiltonian systems that are sufficiently close can be smoothly connected through symplectomorphisms. This concept has deep implications in the study of dynamics and stability in symplectic geometry.
Phase Spaces: Phase spaces are mathematical constructs that represent all possible states of a physical system, including positions and momenta of particles. They provide a framework for analyzing dynamical systems in physics and mathematics, allowing for the visualization and study of system behaviors over time. In symplectic geometry, phase spaces play a crucial role in understanding Hamiltonian mechanics and its applications.
Poincaré's Lemma: Poincaré's Lemma is a fundamental result in differential geometry that states that any closed differential form on a star-shaped domain is exact. This means if a differential form has zero exterior derivative, it can be expressed as the exterior derivative of another form. The lemma plays a crucial role in establishing the relationship between topology and differential forms, which becomes essential in understanding concepts like Gromov's non-squeezing theorem.
Standard symplectic structure: The standard symplectic structure is a canonical example of a symplectic form on the Euclidean space $ ext{R}^{2n}$, expressed as the differential form $$ heta = \sum_{i=1}^{n} dx_i \wedge dy_i$$. This structure serves as the foundational model for understanding symplectic manifolds and provides insight into the geometric and dynamical properties of systems. It is essential for exploring examples of symplectic manifolds and understanding results like Gromov's non-squeezing theorem, which highlights important limitations on the behavior of symplectic embeddings.
Symplectic capacity: Symplectic capacity is a numerical invariant associated with a symplectic manifold that measures the size of a subset in a way that respects the symplectic structure. It plays a significant role in understanding the geometry and topology of symplectic manifolds, providing insights into their properties and behaviors, especially in the context of embedding problems and dynamical systems.
Symplectic Diffeomorphism: A symplectic diffeomorphism is a smooth, invertible mapping between two symplectic manifolds that preserves the symplectic structure, meaning it maintains the differential 2-form associated with the symplectic structure. This concept is crucial because it allows for the comparison and transformation of different symplectic geometries while ensuring that their essential geometric properties remain intact. These mappings play a vital role in various applications, such as Darboux's theorem, group actions, and Gromov's non-squeezing theorem.
Symplectic Form: A symplectic form is a closed, non-degenerate 2-form defined on a differentiable manifold, which provides a geometric framework for the study of Hamiltonian mechanics and symplectic geometry. It plays a crucial role in defining the structure of symplectic manifolds, facilitating the formulation of Hamiltonian dynamics, and providing insights into the conservation laws in integrable systems.
Symplectic topology: Symplectic topology is a branch of mathematics that studies the geometric structures and properties of symplectic manifolds, which are smooth manifolds equipped with a closed, non-degenerate 2-form. This field connects deeply with various areas such as Hamiltonian mechanics, the study of dynamical systems, and algebraic geometry, providing tools to understand the shape and behavior of these manifolds under different transformations.
Viktor Guillemin: Viktor Guillemin is a mathematician known for his significant contributions to symplectic geometry, particularly in relation to the study of symplectic capacities and their properties. His work has been crucial in understanding how symplectic capacities can be used to describe the geometric and topological features of symplectic manifolds, as well as their implications for Gromov's non-squeezing theorem.
Volume Preservation: Volume preservation refers to the property of a transformation or flow that maintains the volume of a given region in space. This concept is crucial in symplectic geometry, where certain transformations, such as symplectomorphisms, preserve a symplectic structure, and thus the volume defined by this structure. Volume preservation helps us understand the behavior of dynamical systems and is essential in proofs related to important theorems like Darboux's theorem and Gromov's non-squeezing theorem.