Abstract This paper is devoted to a novel and nontrivial exploration of eight aspects of the geometrical logarithmic capacitance (a very key notion in mathematical physics, quasiconformal geometry and variational calculus) through: (1) identifying with the reduced conformal module; (2) evaluating the minimal log-potential energy; (3) relating to both the volume-radius and the surface-radius; (4) linking with the n-harmonic radius and the log-capacity of the Kevin image of a compact surface; (5) finding the Minkowski inequality and the general variational formula for the log-capacity; (6) pinching the log-isocapacitary inequality from left and solving the left-prescribed problem for the normalized log-capacitary curvature measure; (7) pinching the log-isocapacitary inequality from right and handling the right-prescribed problem for the normalized log-capacitary curvature measure; (8) handling an overdetermination of the n-equilibrium potential of a given convex body via the log-capacitary concavity.

中文翻译：

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几何对数电容

摘要 本文致力于通过以下方式对几何对数电容（数学物理、拟共形几何和变分微积分中的一个非常关键的概念）的八个方面进行新颖而重要的探索：（1）识别简化的共形模；(2) 评估最小对数势能；(3) 与体积半径和表面半径有关；(4) 与n-谐波半径和致密表面凯文图像的对数容量相关联；(5) 求对数容量的 Minkowski 不等式和一般变分公式；(6) 从左边收缩对数等电容不等式，求解归一化对数电容曲率测度的左规定问题；(7) 从右截断对数等容不等式，处理归一化对数电容曲率测度的右规定问题；(8) 通过对数电容凹度处理给定凸体的 n 平衡势的超定。