关于三纯映射的Compactness结果
Compactness results for triholomorphic maps
摘要：
考虑从几乎超Hermite流形$\mathcal{M}^{4m}$到超Kähler流形$\mathcal{N}^{4n}$的三纯映射.这意味着$u \in W^{1,2}$满足四元数德尔巴方程。我们假设$u$在$W^{1,2}$中通过光滑映射是局部强逼近的：那么这样的映射几乎是平稳调和的（当$\mathcal{M}$也是超Kähler时，那么是平稳调和的）。我们表明，在这种更一般的情况下，经典的$ε$ -正则性结果仍然成立。然后，我们解决了一个弱收敛序列$u_\ell \rightharpoonup u_\infty$的强逼近的三全纯映射$u_\ell:\mathcal{M} \to \mathcal{N}$一致有界Dirichlet能量的紧性问题。爆破分析导致，在通常的固定设置，存在一个整流爆破集$Σ$的余维$2$，远离该序列强烈收敛。缺陷措施$Θ(x) {\mathcal{H}}^{4m-2} \llcorner Σ$编码的能量损失的限制，我们证明，对于a.e. $Σ$上的点，$Θ$的值由（有限）数量的光滑非恒定全纯气泡的能量之和给出（这里全纯性被理解为关于. $\mathcal{N}$上的复杂结构，其取决于$Σ$上的选定点）。在$\mathcal{M}$是hyperKähler的情况下，C. Y. Wang（2003）用不同的证明;我们依赖于洛伦兹空间估计。通过校准和同调论证，我们进一步证明了对于包含在Lipschitz图中的$Σ\setminus \text{Sing}_{u_\infty}$的每一部分，我们都能找到唯一的alm.在$\mathcal{M}$上的复合物，其使得该部分伪全纯且光滑，其中$Θ$恒定;此外，起源于这种光滑片的点处的气泡对于常见的复杂结构是全纯的。
英文摘要：
We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperKähler 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 hyperKähler as well, then stationary harmonic). We show that in this more general situation the classical $ε$-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 $Σ$ of codimension $2$, away from which the sequence converges strongly. The defect measure $Θ(x) {\mathcal{H}}^{4m-2} \llcorner Σ$ encodes the loss of energy in the limit; we prove that for a.e. point on $Σ$ the value of $Θ$ 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 $Σ$). In the case that $\mathcal{M}$ is hyperKähler 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 $Σ\setminus \text{Sing}_{u_\infty}$ contained in a Lipschitz graph we find a unique alm. compl. st. on $\mathcal{M}$ that makes the portion pseudoholomorphic and smooth, with $Θ$ constant; moreover the bubbles originating at points of such a smooth piece are holomorphic for a common complex structure.
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