In this paper we consider closed $\text{SL}(3,\mathbb{C})$-structures which are either mean convex or tamed by a symplectic form. These notions were introduced by Donaldson in relation to ${\text{G}}_2$-manifolds with boundary. In particular, we classify nilmanifolds which carry an invariant mean convex closed $\text{SL}(3,\mathbb{C})$-structure and those which admit an invariant mean convex half-flat $\text{SU}(3)$-structure. We also prove that, if a solvmanifold admits an invariant tamed closed $\text{SL}(3,\mathbb{C})$-structure, then it also has an invariant symplectic half-flat $\text{SU}(3)$-structure.

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关闭 $\text{SL}（3，\mathbb{C}）$-nilmanifolds上的结构

本文考虑封闭 $\text{SL}（3，\mathbb{C}）$-平均凸或被辛形式驯服的结构。这些概念由唐纳森（Donaldson）引入${\text{G}}_{\mathrm{2个}}$-具有边界的流形。尤其是，我们对带有不变均值凸封闭的nilmanifolds进行分类$\text{SL}（3，\mathbb{C}）$结构以及那些允许不变的平均凸半平面的结构 $\text{苏}（3）$-结构体。我们还证明，如果一个流形允许一个不变的驯服封闭$\text{SL}（3，\mathbb{C}）$结构，那么它也具有不变的辛半平坦 $\text{苏}（3）$-结构体。