Compactness results for triholomorphic maps

Abstract

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 $\epsilon$-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 C. Y. Wang [41] 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 $\mathcal{N}$.