Let X be a compact Kähler manifold and D be a simple normal crossing divisor on X such that \(K_X+D\) is big and nef. We first prove that the singular Kähler–Einstein metric constructed by Berman–Guenancia is almost-complete on \(X \backslash D\) in the sense of Tian–Yau. In our second main result, we establish the weak convergence of conic Kähler–Einstein metrics of negative curvature to the above-mentioned metric when \(K_X+D\) is merely big, answering partly a recent question posed by Biquard–Guenancia. Potentials of low energy play an important role in our approach.
