arXiv2026
Let $Σ_0\,=\,Σ\setminus\{z_1,\,\ldots,\, z_n\}$ be a finite-type hyperbolic Riemann surface, with its universal cover being identified with $\mathbb H$. We prove that a finite-energy equivariant harmonic map \[ f\colon\mathbb H\;\longrightarrow\;\mathbb{CH}^2 \] is fixed-lift proper at every puncture --- that is, its restriction to a fixed lifted cusp eventually leaves every compact subset of the target --- if and only if its peripheral holonomy is parabolic. For conformal immersions, parabolic holonomy at every puncture also implies completeness of the descended induced metric. We further show that if a polystable parabolic $\mathrm{PU}(2,\,1)$-Higgs bundle has local data $α_p\,=\,s_p\,=\,0$ and $Y_p\,\ne\,0$, and if the associated harmonic map is weakly conformal, then the corresponding end is unbranched, complete, and has finite energy. We also show that, for mixed conformal data, stability is equivalent to two explicit parabolic slope inequalities. Finally, for every $n\;\ge\;5$, we construct explicit Higgs data producing mixed unbranched $\mathbb{CH}^2$-$n$-noids with nonidentity unipotent peripheral monodromy, finite total energy, complete induced metric, and fixed-lift proper ends.