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.
Categories: math.CV