Rigidity Theorems by Capacities and Kernels - Texas A&M University (TAMU) Scholar

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 Szeg kernel, the higher-order Bergman kernels, and the sublevel sets of the Greens function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture.

Rigidity Theorems by Capacities and Kernels Academic Article