Stabilized symplectic embeddings of higher-dimensional ellipsoids

TL;DR

The paper advances stabilized high-dimensional ellipsoid embedding theory by linking lower bounds for stabilised embedding capacities to the Lagrangian capacity, with the central bound $\mathcal{EC}_{n-k}(x_2,\dots,x_n) \ge (k+1) C_{\mathrm{Lag}}(E^{2n}(1,x_2,\dots,x_n))$ and the threshold $c_k$ growing to infinity. A key novelty is converting the embedding problem into a Lagrangian-classification framework in $\mathbb{CP}^k\times \mathbb{C}^{n-k}$ and employing neck-stretching to produce rigid holomorphic curve counts that obstruct embeddings. The analysis combines holomorphic curve counts, Borman–Sheridan class obstructions, and gravitational descendants (via Tonkonog) to derive sharp counts, including the $(k-1)!$ count for certain holomorphic planes and consequential two-level buildings. For the case $k=2$, the authors develop a complementary strategy using Lagrangian torus classification and Landau–Ginzburg potentials, yielding exotic monotone tori and associated obstructions. Together, these methods yield new quantitative obstructions to stabilized ellipsoid embeddings in higher dimensions and illuminate the interaction between symplectic embedding problems and enumerative invariants.

Abstract

We provide a lower bound for the embedding capacity of higher-dimensional symplectic ellipsoids, formulated in terms of the Lagrangian capacity of ellipsoids. Our approach relies on examining the Borman--Sheridan class of a Weinstein neighborhood of a suitable monotone Lagrangian torus, using Tonkonog's string topology-based computation of the gravitational descendants of the torus.

Theorems & Definitions (14)

• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Definition 2.1
• Theorem 2.2: Gutt--Hutchings MR3868228
• Theorem 2.3: MR4332489
• Theorem 2.4
• Definition 2.5