Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness results for triholomorphic maps   We consider triholomorphic maps from an almost hyper-Hermitian manifold
$\mathcal{M}^{4m}$ into a hyperK\"ahler manifold $\mathcal{N}^{4n}$. This means
that $u \in W^{1,2}$ 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 (when $\mathcal{M}$ is
hyperK\"ahler as well, then stationary harmonic). We show that in this more
general situation the classical $\epsilon$-regularity result still holds. 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 $\Theta(x)
{\mathcal{H}}^{4m-2} \llcorner \Sigma$ encodes the loss of energy in the limit;
we prove that for a.e. point on $\Sigma$ the value of $\Theta$ is given by the
sum of energies of a (finite) number of smooth non-constant holomorphic bubbles
(here the holomorphicity is understood w.r.t. a complex structure on
$\mathcal{N}$ that depends on the chosen point on $\Sigma$). In the case that
$\mathcal{M}$ is hyperK\"ahler this result was established by C. Y. Wang (2003)
with a different proof; we rely on Lorentz space estimates. By means of a
calibration and a homological argument we further prove that for each portion
of $\Sigma \setminus \text{Sing}_{u_\infty}$ contained in a Lipschitz graph we
find a unique alm. compl. st. on $\mathcal{M}$ that makes the portion
pseudoholomorphic and smooth, with $\Theta$ constant; moreover the bubbles
originating at points of such a smooth piece are holomorphic for a common
complex structure.

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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga