The ellipsoid embedding function of a symplectic four-manifold measures the amount
by which its symplectic form must be scaled in order for it to admit an embedding of
an ellipsoid of varying eccentricity. This function generalizes the Gromov width and
ball packing numbers. In the one continuous family of symplectic four-manifolds
that has been analyzed, one-point blowups of the complex projective plane,
there is an open dense set of symplectic forms whose ellipsoid embedding
functions are completely described by finitely many obstructions, while there is
simultaneously a Cantor set of symplectic forms for which an infinite number
of obstructions are needed. In the latter case, we say that the embedding
function has an infinite staircase. In this paper we identify a new infinite
staircase when the target is a four-dimensional polydisk, extending a countable
family identified by Usher (2019). Our work computes the function on infinitely
many intervals and thereby indicates a method of proof for a conjecture of
Usher.
Keywords
symplectic embeddings in four dimensions, ellipsoid
embedding capacity function, infinite staircases, continued
fractions, quantitative symplectic geometry, polydisks