Bellettini, C; Tian, G; (2019) Compactness results for triholomorphic maps. Journal of the European Mathematical Society , 21 (5) pp. 1271-1317. 10.4171/JEMS/860.

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Abstract

We consider triholomorphic maps from an almost hyper-Hermitian manifold M4m into a (simply connected) hyperKähler manifold N4n. This notion entails that the map u∈W1,2 satisfies a quaternionic del-bar equation. We work under the assumption that u is locally strongly approximable in W1,2 by smooth maps: then such maps are almost stationary harmonic, in a suitable sense (in the important special case that M is hyperKähler as well, then they are stationary harmonic). We show, by means of the bmo-h1-duality, that in this more general situation the classical ε-regularity result still holds and we establish the validity, for triholomorphic maps, of the W2,1-conjecture (i.e. an a priori W2,1-estimate in terms of the energy). We then address compactness issues for a weakly converging sequence uℓ⇀u∞ of strongly approximable triholomorphic maps uℓ:M→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 Σ of codimension 2, away from which the sequence converges strongly. The defect measure Θ(x)H4m−2⌊Σ encodes the loss of energy in the limit and we prove that for a.e. point on Σ the value of Θ 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 N that depends on the chosen point on Σ). In the case that 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 Σ∩(M∖Singu∞) to an open set is covered by a Lipschitz connected graph, then actually this portion of Σ is a smooth submanifold without boundary and it is pseudo-holomorphic for a (unique) almost complex structure on M (with Θ constant on this portion); moreover the bubbles originating at points of such a smooth piece are all holomorphic for a common complex structure on N.

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