Obstruction to Symplectic Embeddings between Toric Domains
Abstract
We study symplectic embeddings between four-dimensional toric domains using embedded contact homology. Extending Hutchings' criterion for embeddings between convex toric domains, we obtain obstructions for embeddings from convex toric domains into concave toric domains, from semi-weakly convex toric domains into convex toric domains, and from concave toric domains into concave toric domains. The obstructions are formulated in terms of factorizations of convex, concave, and semi-weakly convex generators together with combinatorial constraints relating their ECH indices and actions. As applications, we recover the sharp obstruction for symplectic embeddings of a polydisk into a union of cylinders, and obtain sharp results for embeddings of certain quadrilateral toric domains into balls and ellipsoids. We also show that these obstructions are in some cases strictly stronger than those coming only from ECH capacities.
Related
- Obstructions in Symplectic Embedding Problems for Four-Dimensional Convex Toric Domains
- Constructing Lagrangian surfaces in all symplectic toric $4$-manifolds from multi-grid diagrams
- Iterated toric descent obstruction and homogeneous spaces over geometric global fields
- Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings
- Decompositions and diagrams of symplectic surfaces in Weinstein domains
BibTeX
@misc{miranda2026obstruction,
title = {{Obstruction to Symplectic Embeddings between Toric Domains}},
author = {Marcelo Miranda and Pedro A. S. Salomão and Jonathan Trejos},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27905},
eprint = {2608.27905},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27905},
}
RIS
TY - UNPB
TI - Obstruction to Symplectic Embeddings between Toric Domains
AU - Marcelo Miranda
AU - Pedro A. S. Salomão
AU - Jonathan Trejos
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.27905
UR - https://arxiv.org/abs/2608.27905
AB - We study symplectic embeddings between four-dimensional toric domains using embedded contact homology. Extending Hutchings' criterion for embeddings between convex toric domains, we obtain obstructions for embeddings from convex toric domains into concave toric domains, from semi-weakly convex toric domains into convex toric domains, and from concave toric domains into concave toric domains. The obstructions are formulated in terms of factorizations of convex, concave, and semi-weakly convex generators together with combinatorial constraints relating their ECH indices and actions. As applications, we recover the sharp obstruction for symplectic embeddings of a polydisk into a union of cylinders, and obtain sharp results for embeddings of certain quadrilateral toric domains into balls and ellipsoids. We also show that these obstructions are in some cases strictly stronger than those coming only from ECH capacities.
ER -