The Dirichlet principle for inner variations

Abstract

We are concerned with the Dirichlet energy of mappings defined on domains in the complex plane. The Dirichlet Principle, the name coined by Riemann, tells us that the outer variation of a harmonic mapping increases its energy. Surprisingly, when one jumps into details about inner variations, which are just a change of independent variables, new equations and related questions start to matter. The inner variational equation, called the Hopf–Laplace equation, is no longer the Laplace equation. Its solutions are generally not harmonic; we refer to them as Hopf harmonics. The natural question that arises is how does a change of variables in the domain of a Hopf harmonic map affect its energy? We show, among other results, that in case of a simply connected domain the energy increases. This should be viewed as Riemann’s Dirichlet Principle for Hopf harmonics. The Dirichlet Principle for Hopf harmonics in domains of higher connectivity is not completely solved. What complicates the matter is the insufficient knowledge of global structure of trajectories of the associated Hopf quadratic differentials, mainly because of the presence of recurrent trajectories. Nevertheless, we have established the Dirichlet Principle whenever the Hopf differential admits closed trajectories and crosscuts. Regardless of these assumptions, we established the so-called Infinitesimal Dirichlet Principle for all domains and all Hopf harmonics. Precisely, the second order term of inner variation of a Hopf harmonic map is always nonnegative.