2018/11/09 by Jin, Alvin, Lee, Andrew S.
#FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1811.03756
We consider the embedding function cb(a) describing the problem of symplectically embedding an ellipsoid E(1,a) into the smallest scaling of the polydisc P(1,b). Previous work suggests that determining the entirety of cb(a) for all b is difficult, as infinite staircases can appear for many sequences of irrational b. In contrast, we show that for every polydisc P(1,b) with b>2, there is an explicit formula for the minimum a such that the embedding problem is determined only by volume. That is, when the ellipsoid is sufficiently stretched, there is a symplectic embedding of E(1,a) fully filling an appropriately scaled polydisc P(λ,λb). Denoted RF(b), this rigid-flexible (RF) value is piecewise smooth with a discrete set of discontinuities for b>2. At the same time, by exhibiting a sequence of obstructive classes for bn = (n+1)/(n) at a=8, we show % that cbn(8) is above the volume constraint. So, in combination with the Frenkel-Müller result, it follows that RF is also discontinuous at b=1.