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Symplectic Weyl Laws

Dan Cristofaro-Gardiner

TL;DR

This work surveys Weyl-type laws in low-dimensional symplectic geometry, centering on how embedded contact homology (ECH) capacities and spectral invariants encode quantitative symplectic size and dynamical information. It develops the foundational Weyl law for ECH capacities on Liouville domains, connects it to Reeb dynamics through the Weinstein conjecture and closing lemmas, and then extends the narrative to surfaces via link spectral invariants and periodic Floer homology (PFH), including proofs of simplicity and rigidity results. The survey then highlights more recent developments, such as elementary spectral invariants, subleading asymptotics, and new normal subgroups, along with topological invariance results for helicity. Throughout, the authors illustrate how Weyl laws yield powerful obstructions, packing results, and dynamical conclusions, with broad implications for symplectic embeddings, boundary phenomena, and global dynamics.

Abstract

We survey a number of Weyl type laws that have recently been established in low-dimensional symplectic geometry. These have had a number of applications, which we also introduce. We sketch a number of proofs so that the reader can get a sense of how these formulas are proved and how they can be applied.

Symplectic Weyl Laws

TL;DR

This work surveys Weyl-type laws in low-dimensional symplectic geometry, centering on how embedded contact homology (ECH) capacities and spectral invariants encode quantitative symplectic size and dynamical information. It develops the foundational Weyl law for ECH capacities on Liouville domains, connects it to Reeb dynamics through the Weinstein conjecture and closing lemmas, and then extends the narrative to surfaces via link spectral invariants and periodic Floer homology (PFH), including proofs of simplicity and rigidity results. The survey then highlights more recent developments, such as elementary spectral invariants, subleading asymptotics, and new normal subgroups, along with topological invariance results for helicity. Throughout, the authors illustrate how Weyl laws yield powerful obstructions, packing results, and dynamical conclusions, with broad implications for symplectic embeddings, boundary phenomena, and global dynamics.

Abstract

We survey a number of Weyl type laws that have recently been established in low-dimensional symplectic geometry. These have had a number of applications, which we also introduce. We sketch a number of proofs so that the reader can get a sense of how these formulas are proved and how they can be applied.

Paper Structure

This paper contains 32 sections, 21 theorems, 37 equations.

Table of Contents

  1. Introduction.
  2. Outline of the article
  3. The first symplectic Weyl laws and some of their applications.
  4. Symplectic capacity theory.
  5. ECH capacities.
  6. The ECH capacity Weyl law.
  7. Embedded Contact Homology
  8. ECH spectral invariants, the $U$-map, and the definition of ECH capacities
  9. A Weyl law for contact three-manifolds that implies Theorem \ref{['thm:cghr']}
  10. The Weinstein conjecture and the theorem on two orbits
  11. Hidden boundaries
  12. The closing lemma
  13. Low action curves of controlled topology
  14. The proof of the contact Weyl law
  15. Weyl laws for surfaces and their applications
  16. ...and 17 more sections

Key Result

Theorem 2.1

gromov There is a symplectic embedding $B_n(r) {\stackrel{s}\hookrightarrow} Z_n(R)$ if and only if $r \le R.$

Theorems & Definitions (50)

  • Remark 1.1
  • Theorem 2.1
  • Example 2.2
  • Proposition 2.3
  • Example 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proof 1: Sketch of proof
  • Proof 2: Sketch of Scaling and Monotonicity
  • Proof 3: Sketch of proof of Proposition \ref{['prop:echell']}
  • ...and 40 more