Compactness results for triholomorphic maps

Abstract

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperK\"ahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ 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 (when $\mathcal{M}$ is hyperK\"ahler as well, then stationary harmonic). We show that in this more general situation the classical $\epsilon$-regularity result still holds. 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 $\Theta(x) {\mathcal{H}}^{4m-2} \llcorner \Sigma$ encodes the loss of energy in the limit; we prove that for a.e. point on $\Sigma$ the value of $\Theta$ is given by the sum of energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphicity is understood w.r.t. a complex structure on $\mathcal{N}$ that depends on the chosen point on $\Sigma$). In the case that $\mathcal{M}$ is hyperK\"ahler this result was established by C. Y. Wang (2003) with a different proof; we rely on Lorentz space estimates. By means of a calibration and a homological argument we further prove that for each portion of $\Sigma \setminus \text{Sing}_{u_\infty}$ contained in a Lipschitz graph we find a unique alm. compl. st. on $\mathcal{M}$ that makes the portion pseudoholomorphic and smooth, with $\Theta$ constant; moreover the bubbles originating at points of such a smooth piece are holomorphic for a common complex structure.