JournalsjemsVol. 21, No. 5pp. 1271–1317

Compactness results for triholomorphic maps

  • Costante Bellettini

    University of Cambridge, UK
  • Gang Tian

    Princeton University, USA and Peking University, Beijing, China
Compactness results for triholomorphic maps cover

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We consider triholomorphic maps from an almost hyper-Hermitian manifold M4m\mathcal M^{4m} into a (simply connected) hyperKähler manifold N4n\mathcal N^{4n}. This notion entails that the map uW1,2u \in W^{1,2} satisfies a quaternionic del-bar equation. We work under the assumption that uu is locally strongly approximable in W1,2W^{1,2} by smooth maps: then such maps are almost stationary harmonic, in a suitable sense (in the important special case that M\mathcal M is hyperKähler as well, then they are stationary harmonic). We show, by means of the bmo-h1\mathscr{h}^1-duality, that in this more general situation the classical ε\varepsilon-regularity result still holds and we establish the validity, for triholomorphic maps, of the W2,1W^{2,1}-conjecture (i.e. an a priori W2,1W^{2,1}-estimate in terms of the energy). We then address compactness issues for a weakly converging sequence uuu_\ell \rightharpoonup u_\infty of strongly approximable triholomorphic maps u:MNu_\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 Σ\Sigma of codimension 2, away from which the sequence converges strongly. The defect measure Θ(x)H4m2Σ\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 Σ\Sigma the value of Θ\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 N\mathcal N that depends on the chosen point on Σ\Sigma). In the case that M\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 Σ(MSingu)\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 Σ\Sigma is a smooth submanifold without boundary and it is pseudo-holomorphic for a (unique) almost complex structure on M\mathcal M (with Θ\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 N\mathcal N.

Cite this article

Costante Bellettini, Gang Tian, Compactness results for triholomorphic maps. J. Eur. Math. Soc. 21 (2019), no. 5, pp. 1271–1317

DOI 10.4171/JEMS/860