Rigidity theorems by capacities and kernels

For any open hyperbolic Riemann surface \(X\), the Bergman kernel \(K\), the logarithmic capacity \(c_{\beta}\), and the analytic capacity \(c_{B}\) satisfy the inequality chain \(\pi K \geq c^2_{\beta} \geq c^2_B\); moreover, equality holds at a single point between any two of the three quantities if and only if \(X\) is biholomorphic to a disk possibly less a relatively closed polar set. We extend the inequality chain by showing that \(c_{B}^2 \geq \pi v^{-1}(X)\) on planar domains, where \(v(\cdot)\) is the Euclidean volume, and characterize the extremal cases when equality holds at one point. Similar rigidity theorems concerning the Szeg\"{o} kernel, the higher-order Bergman kernels, and the sublevel sets of the Green's function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture.