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.

我们关注在复平面上的域上定义的映射的Dirichlet能量。黎曼（Dirichlet Principle）是里曼（Riemann）创造的名字，它告诉我们谐波映射的外部变化会增加其能量。令人惊讶的是，当人们跳进内部变化的细节时，这仅仅是自变量的变化，新的方程式和相关的问题开始变得重要。称为Hopf–Laplace方程的内部变分方程不再是Laplace方程。它的解决方案通常不是谐波。我们称它们为Hopf谐波。随之而来的自然问题是，霍普夫谐波图域中的变量变化如何影响其能量？我们显示，除其他结果外，在简单连接的域中，能量会增加。这应该被视为霍普夫谐波的黎曼狄利克雷原理。连通性较高的域中的Hopf谐波的Dirichlet原理尚未完全解决。使问题复杂化的是，对相关联的Hopf二次微分的轨迹的整体结构的了解不足，主要是因为存在重复轨迹。尽管如此，只要霍普夫差速器允许闭合轨迹和横切面，我们就建立了狄利克雷原理。无论这些假设如何，我们都建立了所谓的无穷小Dirichlet原理适用于所有域和所有Hopf谐波。准确地说，霍普夫谐波图的内部变化的二阶项始终为负。