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 v₁, ..., v_2k there is

[math]\displaystyle{ (\underbrace{\omega\wedge\cdots\wedge\omega}_{k\text{ times}})(v_1,\ldots,v_{2k}) \leq k !. }[/math]

There is equality if and only if the span of v₁, ..., v_2k is closed under the operation of J.({{{1}}}, {{{2}}})

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!.({{{1}}}, {{{2}}})

Proof

k = 1

In the special case k = 1, the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality:

[math]\displaystyle{ \omega(v_1,v_2)=g(J(v_1),v_2)\leq \|J(v_1)\|_g\|v_2\|_g=1. }[/math]

According to the equality case of the Cauchy–Schwarz inequality, equality occurs if and only if J(v₁) and v₂ are collinear, which is equivalent to the span of v₁, v₂ being closed under J.

k > 1

Let v₁, ..., v_2k be fixed, and let T denote their span. Then there is an orthonormal basis e₁, ..., e_2k of T with dual basis w₁, ..., w_2k such that

[math]\displaystyle{ \iota^\ast\omega=\sum_{j=1}^k\omega(e_{2j-1},e_{2j})w_{2j-1}\wedge w_{2j}, }[/math]

where ι denotes the inclusion map from T into V.({{{1}}}, {{{2}}}) This implies

[math]\displaystyle{ \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}, }[/math]

which in turn implies

[math]\displaystyle{ (\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!, }[/math]

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 e₁, ..., e_2k is closed under J.

Finally, the dependence of the quantity

[math]\displaystyle{ (\underbrace{\omega\wedge\cdots\wedge\omega}_{k\text{ times}})(v_1,\ldots,v_{2k}) }[/math]

on v₁, ..., v_2k is only on the quantity v₁ ∧ ⋅⋅⋅ ∧ v_2k, and from the orthonormality condition on v₁, ..., v_2k, this wedge product is well-determined up to a sign. This relates the above work with e₁, ..., e_2k to the desired statement in terms of v₁, ..., 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

[math]\displaystyle{ \operatorname{vol}(M)\geq\frac{1}{k!}\int_M \omega^k, }[/math]

where ω is the Kähler form of the metric. Furthermore, equality is achieved if and only if M is a complex submanifold.({{{1}}}, {{{2}}}) 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.({{{1}}}, {{{2}}}) 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.({{{1}}}, {{{2}}})

See also

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

References

• Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7. 
• Griffiths, Phillip; Harris, Joseph (1978). Principles of algebraic geometry. Pure and Applied Mathematics. New York: John Wiley & Sons. ISBN 0-471-32792-1. 
• Harvey, Reese; Lawson, H. Blaine Jr. (1982). "Calibrated geometries". Acta Mathematica 148: 47–157. doi:10.1007/BF02392726. 
• McDuff, Dusa; Salamon, Dietmar (2017). Introduction to symplectic topology. Oxford Graduate Texts in Mathematics (Third edition of 1995 original ed.). Oxford: Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN 978-0-19-879490-5. 
• Wirtinger, W. (1936). "Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde in euklidischer und Hermitescher Maßbestimmung". Monatshefte für Mathematik und Physik 44: 343–365. doi:10.1007/BF01699328.