Teichmüller’s Modulsatz and the Variation of the Dirichlet Integral

Abstract

We show thatchanging the level curve of a harmonic functionwith the classical Hadamard variation with a small parameterentails a change in the Dirichlet integral of the functionwhich is quadratic in the parameter.As a corollary,we supplement the well-known theorem of Teichmüllerabout the sum of moduli of doubly connected domainsinto which an annulus is subdividedby a continuum that differs little from a concentric circle.