Subgraph densities in a surface

Given a fixed graph $H$ that embeds in a surface $\mathrm{\Sigma }$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\mathrm{\Sigma }$? We show that the answer is $\mathrm{\Theta }(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number’ of $H$, which is independent of $\mathrm{\Sigma }$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.