In this article, we present characterizations of the concavity property of minimal $L^2$ integrals degenerating to linearity in the case of products of analytic subsets on products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $L^2$ extension problem from products of analytic subsets to products of open Riemann surfaces, which implies characterizations of the product versions of the equality parts of Suita conjecture and extended Suita conjecture, and the equality holding of a conjecture of Ohsawa for products of open Riemann surfaces.
We prove that if the frequency of the quasi-periodic $\textrm{SL}(2,{{\mathbb {R}}})$ cocycle is Diophantine, then each of the following properties is dense in the subcritical regime: for any $\frac{1}{2}<\kappa <1$, the Lyapunov exponent is exactly $\kappa $-Hölder continuous; the extended eigenstates of the potential have optimal sub-linear growth; and the dual operator associated with a subcritical potential has power-law decaying eigenfunctions. The proof is based on fibered Anosov–Katok constructions for quasi-periodic $\textrm{SL}(2,{{\mathbb {R}}})$ cocycles.
We study the convergence rate of Bergman metrics on the class of polarized pointed Kähler n-manifolds (M, L, g, x) with $\textrm{Vol}\left( B_1 (x) \right) >v $ and $|\!\sec \!|\le K $ on M. Relying on Tian’s peak section method (Tian in J Differ Geom 32(1):99–130, 1990), we show that the $C^{1,\alpha }$ convergence of Bergman metrics is uniform. In the end, we discuss the sharpness of our estimates.