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 M, the exterior kth power of the symplectic form (Kähler form) ω, when evaluated on a simple (decomposable) (2k)-vector ζ of unit volume, is bounded above by k!. That is,

 \omega^k(\zeta) \leq k !\,.

In other words,  \textstyle{\frac{\omega^k}{k!}} is a calibration on  M . An important corollary is that every complex submanifold of a Kähler manifold is volume minimizing in its homology class.

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