Parole chiave

• quadratic differential
• capacity extremal problems
• conformal module

Abstract We will discuss several extremal problems on capacity and conformal module, where quadratic differentials play a significant role. Among those problems are Teichmüller’s problem on the minimal capacity of rings separating pairs of points, Ahlfors’ problem on separating circles, the Squeezing problem, and problems related to Jenkins-Strebel quadratic differentials. We will review known results and present a few new results in the two-dimensional case. Also, we will discuss three-dimensional counterparts of the problems mentioned above. In dimension three, only few results are known and most of these problems remain unsolved and notoriously difficult.