Quantization for biharmonic maps from non-collapsed degenerating Einstein 4-manifolds

For a sequence of extrinsic or intrinsic biharmonic maps $u_j: M_j\rightarrow N$fromasequenceofnon-collapseddegeneratingclosedEinstein4-manifolds$(M_j,g_j)$withboundedEinsteinconstants,boundeddiametersandbounded$L^2$curvatureenergyintoacompactRiemannianmanifold$(N,h)$ with uniformly bounded biharmonic energy, we establish a compactness theory modular finitely many bubbles, which are finite energy biharmonic maps from ${\mathbb{R}}^4$, or from ${\mathbb{R}}^4\mathrm{/}\mathrm{\Gamma }$ for some nontrivial finite group $\mathrm{\Gamma }\subset SO(4)$, or from some complete, noncompact, Ricci flat, non-flat ALE 4-manifold (orbifold). To achieve this, we develop a sophisticated asymptotic analysis for solutions over degenerating neck regions.