Wirtinger inequality (2-forms)
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- For other inequalities named after Wirtinger, see Wirtinger's inequality.
In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold , the exterior
th power of the symplectic form (Kähler form) ω, when evaluated on a simple (decomposable)
-vector ζ of unit volume, is bounded above by
. That is,
In other words, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \textstyle{\frac{\omega^k}{k!}}
is a calibration on. An important corollary is that every complex submanifold of a Kähler manifold is volume minimizing in its homology class.
See also
References
- Victor Bangert; Mikhail Katz; Steve Shnider; Shmuel Weinberger: E_7, Wirtinger inequalities, Cayley 4-form, and homotopy. Duke Math. J. 146 ('09), no. 1, 35-70. See arXiv:math.DG/0608006