Compactness results for triholomorphic maps

  • Costante Bellettini

    University of Cambridge, UK
  • Gang Tian

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

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