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 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,

${\displaystyle \omega ^{k}(\zeta )\leq k!\,.}$

For other inequalities named after Wirtinger, see Wirtinger's inequality.

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.