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: ''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 , the exterior th power of the symplectic form (Kähler form) , when evaluated on a simple (decomposable) -vector of unit volume, is bounded above by . That is, : (\underbrace_)(v_1,\ldots,v_) \leq k ! for any orthonormal vectors . In other words, is a calibration on . 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 In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
* Gromov's inequality for complex projective space *
Systolic geometry In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, ...


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References

*{{cite book, last = Federer, first = Herbert, author-link1=Herbert Federer, title = Geometric measure theory, place= Berlin–Heidelberg–New York, publisher = Springer-Verlag, series = Die Grundlehren der mathematischen Wissenschaften, volume = 153, year = 1969, isbn = 978-3-540-60656-7, mr=0257325, zbl= 0176.00801 , doi=10.1007/978-3-642-62010-2 Inequalities Differential geometry Systolic geometry