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 Template:Math, symplectic form Template:Math, and almost-complex structure Template:Math, linked by Template:Math for any vectors Template:Math and Template:Math. Then for any orthonormal vectors Template:Math there is

$(\underset{k\text{ times}}{\underbrace{\omega \wedge \cdots \wedge \omega }})(v_1,\dots ,v_{2k})\le k!.$

There is equality if and only if the span of Template:Math is closed under the operation of Template:Math.Template:Sfnm

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 Template:Math is equal to Template:Math.Template:Sfnm

Proof

Template:Math

In the special case Template:Math, the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality:

$\omega (v_1,v_2)=g(J(v_1),v_2)\le \Vert J(v_1){\Vert }_g\Vert v_2{\Vert }_g=1.$

According to the equality case of the Cauchy–Schwarz inequality, equality occurs if and only if Template:Math and Template:Math are collinear, which is equivalent to the span of Template:Math being closed under Template:Mvar.

Template:Math

Let Template:Math be fixed, and let Template:Mvar denote their span. Then there is an orthonormal basis Template:Math of Template:Mvar with dual basis Template:Math such that

${\iota }^{\ast }\omega =\sum _{j=1}^k\omega (e_{2j-1},e_{2j})w_{2j-1}\wedge w_{2j},$

where Template:Math denotes the inclusion map from Template:Mvar into Template:Mvar.Template:Sfnm This implies

$\underset{k\text{ times}}{\underbrace{{\iota }^{\ast }\omega \wedge \cdots \wedge {\iota }^{\ast }\omega }}=k!\prod _{i=1}^k\omega (e_{2i-1},e_{2i})w_1\wedge \cdots \wedge w_{2k},$

which in turn implies

$(\underset{k\text{ times}}{\underbrace{\omega \wedge \cdots \wedge \omega }})(e_1,\dots ,e_{2k})=k!\prod _{i=1}^k\omega (e_{2i-1},e_{2i})\le k!,$

where the inequality follows from the previously-established Template:Math case. If equality holds, then according to the Template:Math equality case, it must be the case that Template:Math for each Template:Mvar. This is equivalent to either Template:Math or Template:Math, which in either case (from the Template:Math case) implies that the span of Template:Math is closed under Template:Math, and hence that the span of Template:Math is closed under Template:Mvar.

Finally, the dependence of the quantity

$(\underset{k\text{ times}}{\underbrace{\omega \wedge \cdots \wedge \omega }})(v_1,\dots ,v_{2k})$

on Template:Math is only on the quantity Template:Math, and from the orthonormality condition on Template:Math, this wedge product is well-determined up to a sign. This relates the above work with Template:Math to the desired statement in terms of Template:Math.

Consequences

Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any Template:Math-dimensional embedded submanifold Template:Mvar, there is

$\mathrm{vol}(M)\ge \frac{1}{k!}\underset{M}{\int }{\omega }^k,$

where Template:Math is the Kähler form of the metric. Furthermore, equality is achieved if and only if Template:Mvar is a complex submanifold.Template:Sfnm In the special case that the hermitian metric satisfies the Kähler condition, this says that Template:Math is a calibration for the underlying Riemannian metric, and that the corresponding calibrated submanifolds are the complex submanifolds of complex dimension Template:Mvar.Template:Sfnm 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.Template:Sfnm

See also

• Gromov's inequality for complex projective space
• Systolic geometry

References

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