The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

Neumann-Poincaré (NP) 算子是域边界上的奇异积分算子，在通过边界积分公式解决电导率传输问题时自然会出现。最近，基于几何函数理论 [34] 开发了二维的 NP 算子的级数表达式。在本文中，我们使用该级数展开研究复合材料的几何特性。特别是，根据相关的外部保形映射，我们获得了极化张量和有效电导率的明确公式，用于包含具有极值电导率的任意形状的夹杂物或周期性夹杂物阵列。此外，我们通过数值计算观察到，NP 算子的谱对于夹杂物的形状变形具有单调行为。此外，