# Compactness results for triholomorphic maps

### Costante Bellettini

University of Cambridge, UK### Gang Tian

Princeton University, USA and Peking University, Beijing, China

## Abstract

We consider triholomorphic maps from an almost hyper-Hermitian manifold $M_{4m}$ into a (simply connected) hyperKähler manifold $N_{4n}$. This notion entails that the map $u∈W_{1,2}$ 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 $M$ is hyperKähler as well, then they are stationary harmonic). We show, by means of the bmo-$h_{1}$-duality, that in this more general situation the classical $ε$-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 $u_{ℓ}⇀u_{∞}$ of strongly approximable triholomorphic maps $u_{ℓ}:M→N$ 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 $Θ(x)H_{4m−2}⌊Σ$ 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 $Σ∩(M∖Sing_{u_{∞}})$ 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 $N$.

## 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