The mean curvature of first-order submanifolds in exceptional geometries with torsion
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The mean curvature of first‑order submanifolds in exceptional geometries with torsion Gavin Ball1 · Jesse Madnick2 Received: 6 March 2020 / Accepted: 29 August 2020 © Springer Nature B.V. 2020
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 G2-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 G2-structures with torsion. Keywords Associative · Coassociative · Cayley · Mean curvature · Intrinsic torsion · G-structure
1 Introduction In their fundamental work on calibrations, Harvey and Lawson [10] defined four new classes of calibrated submanifolds in Riemannian manifolds with special holonomy, summarized in the following table: Submanifold
Ambient Manifold
Special Lagrangian n-fold
Riemannian 2n-manifold (M 2n , g) with Hol(g) ≤ SU(n)
Associative 3-fold
Riemannian 7-manifold (M 7 , g) with Hol(g) ≤ G2
Coassociative 4-fold Cayley 4-fold
Riemannian 7-manifold (M 7 , g) with Hol(g) ≤ G2 Riemannian 8-manifold (M 8 , g) with Hol(g) ≤ Spin(7)
By virtue of being calibrated, each of these submanifolds satisfy a strong area-minimizing property. In particular, they are stable minimal submanifolds. Moreover, by * Jesse Madnick [email protected] Gavin Ball [email protected] 1
Département de mathématiques, Université du Québec à Montréal, Case postale 8888, Succursale centre‑ville, Montréal, QC H3C 3P8, Canada
2
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
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Vol.:(0123456789)
Annals of Global Analysis and Geometry
an argument using the Cartan–Kähler Theorem, Harvey and Lawson [10] were able to show that submanifolds of each class exist locally in abundance. Riemannian manifolds with special holonomy groups function as the background spaces for supersymmetric theories of physics. In this setting, calibrated submanifolds are related to supersymmetric cycles [3]. Special Lagrangian submanifolds lie at the foundation of the SYZ formulation of mirror symmetry [14], and calibrated submanifolds in manifolds with holonomy groups G2 and Spin(7) are expected to play a similar role in theories of mirror symmetry for such manifolds [1, 9]. In fact, each of the classes of submanifolds described above make sense in an even more general class of ambient spaces: namely, that of (Riemannian) manifolds M equipped with a G-structure, for G = SU(n) or G2 or Spin(7) as appropriate. Submanifold
Ambient manifold
Special Lagrangian n-fold
2n-manifold M 2n with an SU(n)-structure
Associative 3-fold
7-manifold M 7 with a G2-structure
Coassociative 4-fold
7-manifold M 7 with a G2-structure
Cayley 4-fold
8-manifold M 8 with a Spin(7)-structure
However, in this generalized setting, such submanifolds need
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