Parallel calibrations and minimal submanifolds

2008/08/15 by C. Robles, Robles, C.
Mathematics · #53C38 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C38

paper · pdf · doi:10.48550/arxiv.0808.2158

v2: substantial revision including new result (Theorem 1.2), 10 pages

arxiv created 2009/12/04 · arxiv updated 2009/12/08

Abstract

Given a parallel calibration ϕ∈ Ω^p(M) on a Riemannian manifold M, I prove that the ϕ--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the ϕ--critical submanifolds are precisely the integral manifolds of a \mathscrC^∞(M)--linear subspace \sP ⊂ Ω^p(M). In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi-Yau manifolds, (co)associative calibrations on G₂--manifolds, and the Cayley calibration on \tSpin(7)--manifolds.)

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