Noncontractible loops of symplectic embeddings between convex toric domains (Q2216146)

For \(i=1,2\) let \(f_i:[0,a_i]\to\mathbb{R}_{\geq0}\) be a nonincreasing concave function with \(f_i(0)=b_i\), \(f_i(a_i)=0\). The corresponding convex toric domains are defined by \[ X_i= \left\{(z_1,z_2)\in\mathbb{C}^2:\pi|z_1|^2\leq a_i, \pi|z_2|^2\leq f_i(\pi|z_1|^2)\right\} \] and are equipped with the induced standard symplectic structure. Let \(\Phi_t:\mathbb{C}^2\to\mathbb{C}^2\) be the concatenation of the \(2\pi\) counterclockwise rotation in the \(z_1\)-plane followed by the \(2\pi\) clockwise rotation in the \(z_2\)-plane. This loop is contractible in \(\mathrm{Sp}(4,\mathbb{R})\). The main result of the paper is the following. Let the functions \(f_i(x)\) be what the author called nice, which in particular implies that there exists a unique point \(c_i\in(0,a_i)\) with \(f_i'(c_i)=-1\). Assume that \(X_1\subset X_2\), \(a_1< a_2< b_1< b_2\) and \(a_2<\min(2a_1,c_1+f_1(c_1))\). Then the loop of symplectic embeddings \(\Phi_t|_{X_1}\) is noncontractible among the symplectic embedding of \(X_1\) to \(X_2\). This statement also generalizes to not necessary nice functions and, in particular, applies to symplectic ellipsoids \(E(a_i,b_i)\), corresponding to \(f_i(x)=b_i(1-x/a_i)\), subject to the inequalities \(a_1< a_2< b_1< b_2\), \(a_2<2a_1\). Note that if \(a_1,b_1 < r < a_2,b_2\) for some \(r\) then there is a ball \(B(r)=E(r,r)\) between the two ellipsoids and the loop is contractible.