# Wirtinger inequality (2-forms)

    For other inequalities named after Wirtinger, see Wirtinger's inequality.
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 g, symplectic form ω, and almost-complex structure J, linked by ω(u, v) = g(J(u), v) for any vectors u and v. Then for any orthonormal vectors v$_1$, ..., v$_{2k}$ there is
    $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})\leq k!.$
There is equality if and only if the span of v$_1$, ..., v$_{2k}$ is closed under the operation of J.
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 ω ∧ ⋅⋅⋅ ∧ ω is equal to k!.

## Proof
### k = 1

In the special case k = 1, the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality:
    $\omega (v_{1},v_{2})=g(J(v_{1}),v_{2})\leq \|J(v_{1})\|_{g}\|v_{2}\|_{g}=1.$
According to the equality case of the Cauchy–Schwarz inequality, equality occurs if and only if J(v$_1$) and v$_2$ are collinear, which is equivalent to the span of v$_1$, v$_2$ being closed under J.

### k > 1

Let v$_1$, ..., v$_{2k}$ be fixed, and let T denote their span. Then there is an orthonormal basis e$_1$, ..., e$_{2k}$ of T with dual basis w$_1$, ..., w$_{2k}$ such that
    $\iota ^{\ast }\omega =\sum _{j=1}^{k}\omega (e_{2j-1},e_{2j})w_{2j-1}\wedge w_{2j},$
where ι denotes the inclusion map from T into V. This implies
    $\underbrace {\iota ^{\ast }\omega \wedge \cdots \wedge \iota ^{\ast }\omega } _{k{\text{ times}}}=k!\prod _{i=1}^{k}\omega (e_{2i-1},e_{2i})w_{1}\wedge \cdots \wedge w_{2k},$
which in turn implies
    $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(e_{1},\ldots ,e_{2k})=k!\prod _{i=1}^{k}\omega (e_{2i-1},e_{2i})\leq k!,$
where the inequality follows from the previously-established k = 1 case. If equality holds, then according to the k = 1 equality case, it must be the case that ω(e$_{2i−1}$, e$_{2i}$) = ±1 for each i. This is equivalent to either ω(e$_{2i−1}$, e$_{2i}$) = 1 or ω(e$_{2i}$, e$_{2i−1}$) = 1, which in either case (from the k = 1 case) implies that the span of e$_{2i−1}$, e$_{2i}$ is closed under J, and hence that the span of e$_1$, ..., e$_{2k}$ is closed under J.
Finally, the dependence of the quantity
    $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})$
on v$_1$, ..., v$_{2k}$ is only on the quantity v$_1$ ∧ ⋅⋅⋅ ∧ v$_{2k}$, and from the orthonormality condition on v$_1$, ..., v$_{2k}$, this wedge product is well-determined up to a sign. This relates the above work with e$_1$, ..., e$_{2k}$ to the desired statement in terms of v$_1$, ..., v$_{2k}$.

## Consequences

Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any 2k-dimensional embedded submanifold M, there is
    $\operatorname {vol} (M)\geq {\frac {1}{k!}}\int _{M}\omega ^{k},$
where ω is the Kähler form of the metric. Furthermore, equality is achieved if and only if M is a complex submanifold. In the special case that the hermitian metric satisfies the Kähler condition, this says that ⁠1/k!⁠ω$^k$ is a calibration for the underlying Riemannian metric, and that the corresponding calibrated submanifolds are the complex submanifolds of complex dimension k. 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.