Wirtinger inequality (2-forms)

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.

Statement

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 can also be phrased as saying that the comass of the form is equal to.

Proof

In the special case, the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality:
According to the equality case of the Cauchy–Schwarz 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 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.

Consequences

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.