In mathematics, the Wirtinger inequality, named after Wilhelm Wirtinger, is a fundamental result in complex linear algebra which relates the symplectic and volume forms of a hermitian inner product. It has important consequences in complex geometry, such as showing that the normalized exterior powers of the Kähler form of a Kähler manifold are calibrations.
Consider a real vector space with positive-definite inner product , symplectic form , and almost-complex structure , linked by for any vectors and . Then for any orthonormal vectors there is
There is equality if and only if the span of is closed under the operation of .
In the language of the comass of a form, the Wirtinger theorem (although without precision about when equality is achieved) can also be phrased as saying that the comass of the form is equal to .
====== In the special case , the Wirtinger inequality is a special case of the CauchyâÂÂSchwarz inequality:
According to the equality case of the CauchyâÂÂSchwarz inequality, equality occurs if and only if and are collinear, which is equivalent to the span of being closed under . ====== Let be fixed, and let denote their span. Then there is an orthonormal basis of with dual basis such that
where denotes the inclusion map from into . This implies
which in turn implies
where the inequality follows from the previously-established case. If equality holds, then according to the equality case, it must be the case that for each . This is equivalent to either or , which in either case (from the case) implies that the span of is closed under , and hence that the span of is closed under .
Finally, the dependence of the quantity
on is only on the quantity , and from the orthonormality condition on , this wedge product is well-determined up to a sign. This relates the above work with to the desired statement in terms of .
Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any -dimensional embedded submanifold , there is
where is the Kähler form of the metric. Furthermore, equality is achieved if and only if is a complex submanifold. In the special case that the hermitian metric satisfies the Kähler condition, this says that is a calibration for the underlying Riemannian metric, and that the corresponding calibrated submanifolds are the complex submanifolds of complex dimension . This says in particular that every complex submanifold of a Kähler manifold is a minimal submanifold, and is even volume-minimizing among all submanifolds in its homology class.
Using the Wirtinger inequality, these facts even extend to the more sophisticated context of currents in Kähler manifolds.