Gromov width

Definition

Gromov width is a symplectic capacity that measures the largest size of a symplectic submanifold that can be embedded into a given symplectic manifold. It connects various concepts in symplectic geometry, particularly in terms of how symplectic manifolds can accommodate certain geometric shapes and structures. The Gromov width provides insights into the relationships between different symplectic manifolds and has important implications for understanding their properties and behavior.

5 Must Know Facts For Your Next Test

1. The Gromov width of a symplectic manifold is defined as the supremum of the sizes of all Lagrangian embeddings of an n-dimensional ball into the manifold.
2. It is closely related to other symplectic capacities, such as the Hofer-Zehnder capacity, providing valuable comparisons between different types of capacities.
3. Gromov width can be used to determine whether two symplectic manifolds are deformation equivalent by analyzing their respective capacities.
4. The Gromov width is invariant under Hamiltonian isotopy, meaning that it does not change when applying Hamiltonian deformations to the symplectic structure.
5. Gromov's theorem shows that if a symplectic manifold has sufficiently large Gromov width, it implies that there exist many Lagrangian submanifolds within it.

Review Questions

• How does Gromov width serve as a tool for comparing different symplectic manifolds?
  • Gromov width acts as a numerical invariant that quantifies how much space a given symplectic manifold can accommodate for Lagrangian embeddings. By measuring the largest size of an embedded ball, we can compare two different symplectic manifolds based on their Gromov widths. If one manifold has a larger Gromov width than another, it suggests that it can support larger or more complex Lagrangian submanifolds, thus giving insights into their relative sizes and properties.
• Discuss the significance of Gromov's theorem in relation to Gromov width and its implications for Lagrangian submanifolds.
  • Gromov's theorem is pivotal because it connects the existence of Lagrangian submanifolds to the Gromov width of a symplectic manifold. It states that if the Gromov width is sufficiently large, then there are many Lagrangian submanifolds present. This implies not just geometric richness but also suggests constraints on the topology of the manifold itself. The theorem reinforces the idea that large capacities indicate complex geometric structures within these spaces.
• Evaluate how Gromov width influences our understanding of Hamiltonian dynamics within symplectic manifolds.
  • Gromov width significantly impacts our understanding of Hamiltonian dynamics by providing insights into how systems evolve within symplectic manifolds. Since Gromov width remains invariant under Hamiltonian isotopies, it reflects persistent properties of the system regardless of how it deforms over time. This stability allows researchers to draw conclusions about long-term behavior and possible trajectories of Hamiltonian systems, facilitating predictions about their dynamics based on the underlying geometric structure represented by Gromov width.

Related terms

Symplectic capacity: A numerical invariant associated with symplectic manifolds that measures their size or volume in a certain sense, allowing for comparisons between different symplectic structures.

Lagrangian submanifold: A submanifold of a symplectic manifold that is maximal dimensional and where the symplectic form restricts to zero, playing a critical role in the study of symplectic geometry.

Gromov's theorem: A fundamental result in symplectic geometry stating that the existence of a Lagrangian submanifold in a symplectic manifold leads to significant implications for the topology and geometry of the manifold.