Rigidity theorems by capacities and kernels

TL;DR

This paper establishes new rigidity theorems linking capacities and kernels on hyperbolic Riemann surfaces, characterizing extremal cases and extending classical inequalities to higher dimensions and other kernel functions.

Contribution

It introduces novel inequalities and rigidity results connecting various capacities and kernels, extending classical theorems and characterizing extremal geometric configurations.

Findings

Equality characterizes the disk or polar sets in hyperbolic surfaces.

Extended inequalities for capacities on planar domains.

Rigidity results for Szeg"o kernel and Green's function sublevel sets.

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\"{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.