Ginzburg–Landau global-minimizer conjecture for the constant- and variable-curvature solutions

Let $(\psi\equiv 0,a=a^{b_r})$ be the constant-curvature solution (1.3), let $(\psi,a)$ be the solution (1.7), and let $h_r$ be the relevant Hermitian metric. Let $b_r$ and $\kappa$ be the parameters in the energy $\mathcal{E}(\psi,a,h_r)$, and let the conditions in (1.5) and (1.7) be those assumed in Theorem 1.1. Global-minimizer conjecture. The constant-curvature solution $(\psi\equiv 0,a=a^{b_r})$ is a global minimizer of $\mathcal{E}(\psi,a,h_r)$ if $b_r>\kappa^2$. If $b_r<\kappa^2$, then the solution (1.7) is a global minimizer of $\mathcal{E}(\psi,a,h_r)$. These stronger assertions concern the global energy landscape beyond the preceding local-minimizer conjecture; the paper presents them as expected statements rather than established results.