Genus two trisections are standard

We show that the only closed 4-manifolds admitting genus two trisections are $S^2\times S^2$ and connected sums of $S^1\times S^3$, ${\mathbb{C}\mathbb{P}}^2$, and ${\overline{\mathbb{C}\mathbb{P}}}^2$ with two summands. Moreover, each of these manifolds admits a unique genus two trisection up to diffeomorphism. The proof relies heavily on the combinatorics of genus two Heegaard diagrams of $S^3$. As a corollary, we classify two-component links contained in a genus two Heegaard surface for $S^3$ with a surface-sloped cosmetic Dehn surgery.