THE DIRICHLET PRINCIPLE FOR MEASURED FOLIATIONS

Abstract. We present the minimum norm or Dirichlet principle for measured foliations on a Riemann surface of finite type. In this setting the principle says that if you minimize total energy in a given measure class, you will find a unique representative which is harmonic and represented by the imaginary part of a holomorphic quadratic differential. We include as part of this prin-ciple the notion of extremal length of a measured foliation and the extremal length functional on Teichmüller space. We show that this functional is dif-ferentiable and that its derivative is represented by the unique holomorphic quadratic differential whose heights are equal to the heights of the initially given measured foliation