A striking result of McDuff and Schlenk asserts that in determining when a
four-dimensional symplectic ellipsoid can be symplectically embedded into a
four-dimensional symplectic ball, the answer is governed by an “infinite
staircase” determined by the odd-index Fibonacci numbers and the golden
mean.
There has recently been considerable interest in better understanding this
phenomenon for more general embedding problems. Here we study embeddings of one
four-dimensional symplectic ellipsoid into another, and we show that if the target is
rational, then the infinite staircase phenomenon found by McDuff and Schlenk can
be characterized completely. Specifically, in the rational case, we show that there is
an infinite staircase in precisely three cases: when the target has “eccentricity”
, or
. In
each of these cases, work of Casals and Vianna shows that the corresponding
embeddings can be constructed explicitly using polytope mutation; meanwhile, for
all other eccentricities, the embedding function is given by the classical
volume obstruction, except on finitely many compact intervals, on which it is
linear.
Our work verifies in the special case of ellipsoids a conjecture by Holm,
Mandini, Pires and the author. The case where the target is the ellipsoid
is also
interesting from the point of view of this Cristofaro-Gardiner–Holm–Mandini–Pires
work: the “staircase obstruction” introduced in that work vanishes for this target,
but nevertheless a staircase does not exist. To prove this, we introduce a new
combinatorial technique for understanding the obstruction coming from
embedded contact homology which is applicable in other situations where
the staircase obstruction vanishes, and so is potentially of independent
interest.
Dedicated to my father on the occasion
of his 85th birthday
