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 of unit volume, is bounded above by k!.[1] That is,

${\displaystyle (\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})\leq k!}$

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

for any orthonormal vectors v₁, ..., v_2k. In other words, ω^k/k! is a calibration on M. An important corollary of the further characterization of equality is that every complex submanifold of a Kähler manifold is volume minimizing in its homology class.