The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm–Loewner evolution curves with the Gaussian free field.

中文翻译：

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适形焊接和流线作用在Loewner和Dirichlet能量之间相互作用

Jordan曲线的Loewner能量是其Loewner驱动项的Dirichlet能量。当且仅当曲线为Weil–Petersson拟圆时才是有限的。在本文中，我们描述了在平面中定义的有限Dirichlet能量函数的切割和焊接操作，从而可以根据Dirichlet能量耗散来表达Loewner能量。我们表明，单位矢量场流线的Loewner能量等于谐波扩展绕组的Dirichlet能量。我们还给出了涉及有限Dirichlet能量的复数值函数的标识，该标识同时表示焊接标识和流水线标识。作为应用，我们证明了两个域的等长弧长等距焊接在能量上是亚可加的，并且在简单连接的域中等电位的能量是单调的。