Volumes of open surfaces

A volume of an open surface measures the rate of growth for the number of pluricanonical sections with simple poles at infinity. By Alexeev and Mori, there exists an absolute minimum for the set of positive volumes, with an explicit – but unrealistically small - bound. I will explain a related conjecture due to KollАr and some existing examples. Then I will explain a new candidate for the surface of the smallest volume, found in a joint work with Wenfei Liu.