Anchored symplectic embeddings

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 sharp quantitative criteria 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.