Comparison of symplectic capacities

TL;DR

The paper addresses how relative symplectic capacities compare and how rigidity phenomena transfer to the relative setting. It proves a chain of inequalities $\tilde{c}^{HZ} \le c^S \le c^{SH}$ and derives heaviness-based criteria for the relative almost existence theorem, along with conditions under which $c^{SH}$ coincides with $c_1^{GH}$ under weakened Conley–Zehnder index assumptions. It then establishes $c_1^{GH} \le c^{SH}$ and, when Reeb orbits satisfy a CZ-index bound, equality $c_1^{GH} = c^{SH}$, using Gysin-type exact triangles. Collectively, these results translate spectral invariants and (equivariant) relative symplectic cohomology into concrete obstructions for relative embeddings and the existence of closed characteristics on level sets.

Abstract

In this paper, we compare the symplectic (co)homology capacity with the spectral capacity in the relative case. This result establishes a chain of inequalities of relative symplectic capacities, which is an analogue of the non-relative case. This comparison gives us a criterion for the relative almost existence theorem in terms of heaviness. Also, we investigate a sufficient condition under which the symplectic (co)homology capacity and the first Gutt-Hutchings capacity are equal in both non-relative and relative cases. This condition is less restrictive than the dynamical convexity.

Theorems & Definitions (32)

• Theorem 1.1: bkfgsi
• Theorem 1.2: Theorem \ref{['mcomparison']}
• Theorem 1.3: gg
• Theorem 1.4: msv
• Theorem 1.5: a
• Corollary 1.6
• Theorem 1.7: a
• Theorem 1.8: Theorem \ref{['slight']}
• Corollary 1.9: Corollary \ref{['nonrela']}
• Remark 1.10