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arXiv · 2209.00156

Associative submanifolds in twisted connected sum $G_2$-manifolds

Gorapada Bera
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Abstract

We introduce a method to construct closed rigid associative submanifolds in twisted connected sum $G_2$-manifolds. More precisely, we prove a gluing theorem of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds under a transverse intersection hypothesis. This is analogous to the gluing theorem for $G_2$-instantons introduced in [SW15]. We rephrase the hypothesis in the special cases where the ACyl associative submanifolds are obtained from holomorphic curves or special Lagrangians in ACyl Calabi-Yau $3$-folds, in terms of algebraic-geometric conditions and topological conditions, respectively. This yields many rigid associative submanifolds with new topological types $S^3$, $\mathbf R\mathbf P^3$ or $\mathbf R\mathbf P^3\#\mathbf R\mathbf P^3$.

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Gorapada Bera. 2022-08-31. Associative submanifolds in twisted connected sum $G_2$-manifolds. https://arxiv.org/abs/2209.00156

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