Two degenerate intersections of Clifford tori in $S^2\times S^2$ and $\mathbb{C}P^2$

Abstract

We construct smooth time-periodic Hamiltonians on the monotone manifolds $S^2\times S^2$ and $\mathbb{C}P^2$ for which the Clifford torus meets its Hamiltonian image in exactly two isolated degenerate points. The product example disproves the degenerate Arnol'd-Givental conjecture.