Rigidity Theorems by Capacities and Kernels

Abstract

Abstract 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 Szegö 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.