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 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 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.

See also[]

• 2-form
• Gromov's inequality for complex projective space
• Systolic geometry

Notes[]

References[]

• Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7. MR 0257325. Zbl 0176.00801.