Wirtinger inequality (2-forms)

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 $k$ ^th 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.

References

Victor Bangert; Mikhail Katz; Steve Shnider; Shmuel Weinberger: E_7, Wirtinger inequalities, Cayley 4-form, and homotopy. Duke Math. J. 146 ('09), no. 1, 35-70. See arXiv:math.DG/0608006

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Wirtinger inequality (2-forms)

See also

References