Associative Submanifolds in Twisted Connected Sum G2-Manifolds

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 [29]. 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 $S^{3}$, $\textbf{R}\textbf{P}^{3}$ or $\textbf{R}\textbf{P}^{3}\#\textbf{R}\textbf{P}^{3}$.