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 $&omega;$, and almost-complex structure $J$, linked by $&omega;(u, v) = g(J(u), v)$ 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 $&omega; ∧ ⋅⋅⋅ ∧ &omega;$ is equal to $k!$.

$k = 1$
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 $&iota;$ 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 $&omega;(e_{2i − 1}, e_{2i}) = ±1$ for each $i$. This is equivalent to either $&omega;(e_{2i − 1}, e_{2i}) = 1$ or $&omega;(e_{2i}, e_{2i − 1}) = 1$, 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 $&omega;$ 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 $1⁄k!&omega;^{k}$ 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.