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Minimality of and Local Obstructions to Associative and Coassociative Submanifolds

Banff International Research Station for Mathematical Innovation and Discovery

Seven-manifolds with a $G_2$-structure possess two distinguished classes of submanifolds: associative 3-folds and coassociative 4-folds. When the $G_2$-structure is torsion-free, such submanifolds are minimal (in fact, calibrated) and exist locally. We are led to ask: For which classes of $G_2$-structures is it the case that associative 3-folds (respectively, coassociative 4-folds) are always minimal submanifolds We will answer this by deriving a simple formula for the mean curvature, in the process uncovering new obstructions to the local existence of coassociatives. Time permitting, we will discuss the analogous results for special Lagrangian 3-folds (respectively Cayley 4-folds) in 6-manifolds with $\mathrm{SU}(3)$-structures (respectively 8-manifolds with $\mathrm{Spin}(7)$-structures). This is joint work with Gavin Ball.

Mathematics; Differential geometry; Geometry

video/mp4

Author affiliation: McMaster University

2019-11-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72218

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/