Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture

In this paper we study the degeneration behavior of the norm of the Riemann $\theta$-function in a family of principally polarized abelian varieties over the punctured complex unit disc in terms of the associated polarized real torus. As an application, we obtain the degeneration behavior of the Zhang--Kawazumi invariant $\varphi(M_t)$ of a family of Riemann surfaces $M_t$ in terms of Zhang's invariant $\varphi(\Gamma)$ of the associated metrized reduction graph $\Gamma$. This allows us to deduce a uniform lower bound for the essential minimum of the N\'eron-Tate height on the tautological cycles of any Jacobian variety over a number field.