Rigidity theorems by capacities and kernels

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 $\pi K\ge c_{\beta }^2\ge c_B^2$; 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 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.

27 pages, final version to appear in International Mathematics Research Notices, IMRN. Supersedes and generalizes arXiv: 2011.05273 and arXiv: 2101.01358

References in corpus (2)

• Estimates for the Bergman Kernel and the Multidimensional Suita Conjecture
• On biholomorphisms between bounded quasi-Reinhardt domains