Anchored symplectic embeddings

TL;DR

This paper introduces the concept of anchored symplectic embeddings, combining symplectic embeddings with boundary knot cobordisms, and uses embedded contact homology to establish criteria for their existence in toric domains.

Contribution

It defines anchored symplectic embeddings and provides quantitative criteria for their existence, highlighting cases where embeddings cannot be upgraded without enlarging the target domain.

Findings

Criteria established for anchored symplectic embeddings in toric domains

Examples where embeddings exist but cannot be anchored without enlarging the domain

Application of embedded contact homology techniques

Abstract

Given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an ``anchored symplectic embedding'' to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). We use techniques from embedded contact homology to determine quantitative critera for when anchored symplectic embeddings exist, for many examples of toric domains. In particular we find examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain.