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Pseudoholomorphic curves in nearly Kaehler Manifolds Aslan, Benjamin


The most natural maps into nearly Kaehler manifolds are pseudoholomorphic curves. When taking the cone of a nearly Kaehler manifold one gets a torsion-free $G_2$ manifold, and the cone of a pseudoholomorphic curve will be an associative submanifold. Most of the work on this topic has been done for specific examples of ambient manifolds since compared to pseudoholomorphic curves in symplectic manifolds little is known about them in the general setting. After reviewing the relevant background I will show how holomorphic data can be used to construct integer invariants and examples of these curves, as done by Bryant for $S^6$ and by Xu for $\mathbb C \mathbb P^3$. I will then show how the latter construction can be combined via the Eells-Salamon correspondence with a generalisation of a formula of Friedrich to compute the Euler number of a conformal and harmonic map into $S^4$. In the end, I will present open problems I am working on such as finding new examples of pseudoholomorphic curves.

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