[Paper Review] On the agreement of symplectic capacities in high dimension

Dan Cristofaro‐Gardiner, Richard Hind|arXiv (Cornell University)|Jul 22, 2023

Geometry and complex manifoldsMathematics3 citations

TL;DR

This paper extends the Gutt-Hutchings-Ramos theorem on symplectic capacities in four dimensions to arbitrary dimensions by proving that all normalized symplectic capacities agree on monotone toric domains. The key innovation is a constructive proof of symplectic embeddings of $L$-shaped domains into infinite cylinders in any dimension, resolving a conjecture by Gutt-Pereira-Ramos and enabling the extension of capacity agreement beyond dimension four.

ABSTRACT

A theorem of Gutt-Hutchings-Ramos asserts that all normalized symplectic capacities give the same value for monotone four-dimensional toric domains. We generalize this theorem to arbitrary dimension. The new ingredient in our proof is the construction of symplectic embeddings of "$L$-shaped" domains in any dimension into corresponding infinite cylinders; this resolves a conjecture of Gutt-Pereira-Ramos in the affirmative.

Motivation & Objective

• To generalize the Gutt-Hutchings-Ramos theorem on symplectic capacity agreement from four to arbitrary dimensions.
• To resolve a conjecture by Gutt-Pereira-Ramos regarding symplectic embeddings of $L$-shaped domains into infinite cylinders in higher dimensions.
• To establish that the Gromov width and cylindrical capacity coincide for monotone toric domains, implying capacity uniqueness.
• To provide a new, constructive method for symplectic embeddings in higher dimensions using Hamiltonian diffeomorphisms on complex projective spaces.
• To extend the framework of symplectic reduction and moment maps to construct embeddings in $\mathbb{C}P^{n-1}(S)$ with controlled support.

Proposed method

• Constructs symplectic embeddings of $L$-shaped domains $L(a_1,\dots,a_n)$ into infinite cylinders $Z(r)$ for $r > \sum a_k$ using Hamiltonian diffeomorphisms on $\mathbb{C}P^{n-1}(S)$.
• Reduces the embedding problem to finding a smooth family of Hamiltonian functions on $\mathbb{C}P^{n-1}(S)$ that map the complement of $V_S$ into the complement of $U_S$.
• Uses symplectic reduction to identify the moment map image of $\partial B(S)$ with a triangle $\Delta_S$ in $\mathbb{R}^{n-1}$, enabling toric fiber analysis.
• Employs a product of 1D Hamiltonian diffeomorphisms $\phi_i^t$ on $\mathbb{C}$ such that $\phi_i(D(S-r)) \subset A(a_i, S - \sum_{j \neq i} a_j)$, ensuring area control.
• Applies a cutoff to the Hamiltonian flow to ensure support lies within $B(S)$, enabling extension to $\mathbb{C}P^{n-1}(S)$.
• Verifies that the resulting flow preserves the required domain inclusions via estimates on $\pi|z|^2$ and the triangle inequality.

Experimental results

Research questions

• RQ1 Do all normalized symplectic capacities agree on monotone toric domains in arbitrary dimensions, as they do in dimension four?
• RQ2 Can $L$-shaped domains in $\mathbb{C}^n$ be symplectically embedded into infinite cylinders $Z(r)$ when $r > \sum a_k$?
• RQ3 Is there a higher-dimensional analogue of the pseudoholomorphic curve techniques used in the four-dimensional case?
• RQ4 Can the Gromov width and cylindrical capacity be shown to coincide for monotone toric domains in higher dimensions?
• RQ5 Does the existence of such embeddings imply uniqueness of normalized symplectic capacities across all monotone toric domains?

Key findings

• All normalized symplectic capacities agree on monotone toric domains in any dimension, generalizing the four-dimensional result of Gutt-Hutchings-Ramos.
• An $L$-shaped domain $L(a_1,\dots,a_n)$ admits a symplectic embedding into $Z(r)$ whenever $r > \sum_{k=1}^n a_k$, confirming a conjecture by Gutt-Pereira-Ramos.
• The Gromov width $c_G(X_\Omega)$ and cylindrical capacity $c_Z(X_\Omega)$ coincide for all monotone toric domains $X_\Omega$, implying capacity uniqueness.
• The construction relies on Hamiltonian diffeomorphisms on $\mathbb{C}P^{n-1}(S)$ that map $\overline{B(S-r)}$ into the complement of $U_S$, with support in the affine part.
• The proof uses a product of 1D Hamiltonian flows with controlled area growth, ensuring $\sum \pi|\phi_i^t(z_i)|^2 < S - a_n$ for small $\epsilon$ and $t \in [0,1]$, thus preserving domain inclusions.
• The result holds even in the case $r = \sum a_k$ when combined with Theorem 4.3 of [8], though the strict inequality suffices for the main capacity agreement result.