Compactness results for triholomorphic maps (Q2416375)

Summary: We consider triholomorphic maps from an almost hyper-Hermitian manifold \(\mathcal M^{4m}\) into a (simply connected) hyperKähler manifold \(\mathcal N^{4n}\). This notion entails that the map \(u \in W^{1,2}\) satisfies a quaternionic del-bar equation. We work under the assumption that \(u\) is locally strongly approximable in \(W^{1,2}\) by smooth maps: then such maps are almost stationary harmonic, in a suitable sense (in the important special case that \(\mathcal M\) is hyperKähler as well, then they are stationary harmonic). We show, by means of the bmo-\(h^1\)-duality, that in this more general situation the classical \(\varepsilon\)-regularity result still holds and we establish the validity, for triholomorphic maps, of the \(W^{2,1}\)-conjecture (i.e. an a priori \(W^{2,1}\)-estimate in terms of the energy). We then address compactness issues for a weakly converging sequence \(u_\ell \rightharpoonup u_\infty\) of strongly approximable triholomorphic maps \(u_\ell:\mathcal M \to \mathcal N\) with uniformly bounded Dirichlet energies. The blow up analysis leads, as in the usual stationary setting, to the existence of a rectifiable blow-up set \(\Sigma\) of codimension 2, away from which the sequence converges strongly. The defect measure \(\Theta(x) {\mathcal H}^{4m-2} \lfloor \Sigma\) encodes the loss of energy in the limit and we prove that for a.e. point on \(\Sigma\) the value of \(\Theta\) is given by the sum of the energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphicity is to be understood with respect to a complex structure on \(\mathcal N\) that depends on the chosen point on \(\Sigma$). In the case that \(\mathcal M\) is hyperKähler this quantization result was established by \textit{C. Wang} [Calc. Var. Partial Differ. Equ. 18, No. 2, 145--158 (2003; Zbl 1038.58017)] with a different proof; our arguments rely on Lorentz spaces estimates. By means of a calibration argument and a homological argument we further prove that whenever the restriction of \(\Sigma \cap (\mathcal M \setminus\mathrm{Sing}_{u_\infty})\) to an open set is covered by a Lipschitz connected graph, then actually this portion of \(\Sigma\) is a smooth submanifold without boundary and it is pseudo-holomorphic for a (unique) almost complex structure on \(\mathcal M\) (with \(\Theta\) constant on this portion); moreover the bubbles originating at points of such a smooth piece are all holomorphic for a common complex structure on \(\mathcal N\).