A variational characterization of calibrated submanifolds

Let M be a fixed compact oriented embedded submanifold of a manifold M ¯ . Consider the volume V ( g ¯ ) = ∫ M vol ( M , g ) as a functional of the ambient metric g ¯ on M ¯ , where g = g ¯ M . We show that g ¯ is a critical point of V with respect to a special class of variations of g ¯ , obtained...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 62; no. 6
Main Authors Cheng, Da Rong, Karigiannis, Spiro, Madnick, Jesse
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2023
Springer Nature B.V
Subjects
Analysis
Calculus of Variations and Optimal Control; Optimization
Calibration
Control
Critical point
Manifolds (mathematics)
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
53C42
53C38
35A15
58E20
53C43
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Summary:Let M be a fixed compact oriented embedded submanifold of a manifold M ¯ . Consider the volume V ( g ¯ ) = ∫ M vol ( M , g ) as a functional of the ambient metric g ¯ on M ¯ , where g = g ¯ M . We show that g ¯ is a critical point of V with respect to a special class of variations of g ¯ , obtained by varying a calibration μ on M ¯ in a particular way, if and only if M is calibrated by μ . We do not assume that the calibration is closed. We prove this for almost complex, associative, coassociative, and Cayley calibrations, generalizing earlier work of Arezzo–Sun in the almost Kähler case. The Cayley case turns out to be particularly interesting, as it behaves quite differently from the others. We also apply these results to obtain a variational characterization of Smith maps.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-023-02513-7