$C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces

We investigate the $C^0$-topology of the group of symplectic diffeomorphisms of positive symplectic rational surfaces. For all but a few exceptions, we prove that the group of Hamiltonian diffeomorphisms forms a connected component in the $C^0$-topology. This provides the first nontrivial case in which the group of Hamiltonian diffeomorphisms is known to be $C^0$-closed inside the group of symplectic diffeomorphisms. The key to our approach is to build a bridge between techniques from symplectic mapping class groups and problems in $C^0$-symplectic topology. Via a careful adaptation of tools from $J$-holomorphic foliation and inflation, we establish the necessary $C^0$-distance estimates. We hope that this serves as an example of how these two subfields can interact fruitfully, and also propose several questions arising from this interplay.