Symplectic capacities on surfaces

We classify capacities on the class Symo(2) of connected symplectic surfaces with at most countably many nonplanar ends. To obtain the classification we study diffeomorphism types of surfaces in Symo(2) of infinite genus with nonplanar ends; it turns out that these types are in bijective correspondence with countable successor ordinals of the form ωα · d + 1, where α is an ordinal and d ≥ 0 is an integer. It also turns out that if S1 and S2 are two open surfaces of infinite genera with at most countably many nonplanar ends, then each of the surfaces embeds into the other. Our classification implies that every capacity on the class of symplectic surfaces in Symo(2) of infinite genus differs from the Hofer–Zehnder capacity by a non-negative finite or infinite constant.