Wirtinger inequality (2 forms)

In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold M , the exterior k ^th power of the symplectic form (Kähler form) ω, when evaluated on a simple (decomposable) ( 2 k ) -vector ζ of unit volume, is bounded above by k ! . That is,

In other words, ω 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.