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Journal of the European Mathematical Society

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Volume 21, Issue 5, 2019, pp. 1271–1317
DOI: 10.4171/JEMS/860

Published online: 2019-01-08

Compactness results for triholomorphic maps

Costante Bellettini[1] and Gang Tian[2]

(1) University of Cambridge, UK
(2) Princeton University, USA and Peking University, Beijing, China

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-$\mathscr{h}^1$-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 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 $\mathcal N$.

Keywords: Almost stationary harmonic maps, hyperKähler manifolds, almost hyper-Hermitian manifolds, quantization of Dirichlet energy, bubbling set, regularity properties, Fueter sections

Bellettini Costante, Tian Gang: Compactness results for triholomorphic maps. J. Eur. Math. Soc. 21 (2019), 1271-1317. doi: 10.4171/JEMS/860