Associative submanifolds in twisted connected sum $G_2$-manifolds

TL;DR

This paper presents a new method for constructing closed rigid associative submanifolds in twisted connected sum $G_2$-manifolds using a gluing theorem for asymptotically cylindrical associative submanifolds.

Contribution

It introduces a gluing theorem for ACyl associative submanifolds in $G_2$-manifolds, enabling the construction of new rigid associative submanifolds with diverse topologies.

Findings

Established a gluing theorem for ACyl associative submanifolds.

Constructed many rigid associative submanifolds with new topological types.

Rephrased hypotheses in algebraic-geometric and topological terms for special cases.

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 $\mathbf R\mathbf P^3\#\mathbf R\mathbf P^3$.