Abstract
We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those \(\text{G}_2\)-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative 4-fold, Cayley 4-fold) is a minimal submanifold. In the process, we obtain new obstructions to the local existence of coassociative 4-folds in \(\text{G}_2\)-structures with torsion.
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Acknowledgements
This work has benefited from conversations with Robert Bryant, Jason Lotay, Thomas Madsen, and Alberto Raffero. The second author would also like to thank McKenzie Wang for his guidance and encouragement. The first author thanks the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics for support during the period in which this article was written.
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Ball, G., Madnick, J. The mean curvature of first-order submanifolds in exceptional geometries with torsion. Ann Glob Anal Geom 59, 27–67 (2021). https://doi.org/10.1007/s10455-020-09735-4
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DOI: https://doi.org/10.1007/s10455-020-09735-4