A symplectic rigidity theorem (Q5896297)

Let S be a hypersurface in symplectic space \({\mathbb{R}}^ 4\) with Darboux coordinates \((p_ 1,q_ 1,p_ 2,q_ 2)\), \(C^ 2\)-close to \(S_ 0=\{p_ 1^ 2+q^ 2_ 1=1\}\), all integral curves on S being closed with lengths near 2\(\pi\). 1) If S is strictly convex in some point then S is not weakly convex in any neighbourhood of the whole integral curve through the point. 2) If S is weakly convex everywhere then it can be transformed into \(S_ 1=\{G(p_ 1,q_ 1)=1\}\) by small linear canonical transformation, \(S_ 1\) being near \(S_ 0\). 3) \(S_ 0\) cannot be transformed into a strictly convex hypersurface by \(C^ 2\)- small canonical transformation.