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 $\pi K\ge c_{\beta }^2\ge c_B^2$; 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 $c_B^2\ge \pi v^{-1}(X)$ 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.