Wirtinger inequality (2-forms)

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

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

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

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