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

${\displaystyle (\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})\leq k!.}$

There is equality if and only if the span of v₁, ..., v_2k is closed under the operation of J. ^[1]

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]

Proof

k = 1

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

${\displaystyle \omega (v_{1},v_{2})=g(J(v_{1}),v_{2})\leq \|J(v_{1})\|_{g}\|v_{2}\|_{g}=1.}$

According to the equality case of the Cauchy–Schwarz inequality, equality occurs if and only if J(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

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

where ι denotes the inclusion map from T into V. ^[2] This implies

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

which in turn implies

${\displaystyle (\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(e_{1},\ldots ,e_{2k})=k!\prod _{i=1}^{k}\omega (e_{2i-1},e_{2i})\leq k!,}$

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

${\displaystyle (\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})}$

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

${\displaystyle \operatorname {vol} (M)\geq {\frac {1}{k!}}\int _{M}\omega ^{k},}$

where ω is the Kähler form of the metric. Furthermore, equality is achieved if and only if M is a complex submanifold. ^[3] 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. ^[4] 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. ^[5]

See also

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

Notes

1. Federer 1969, Section 1.8.2.
2. McDuff & Salamon 2017, Lemma 2.4.5.
3. Griffiths & Harris 1978, Section 0.2.
4. Harvey & Lawson 1982.
5. Federer 1969, Section 5.4.19.

References

• Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN   978-3-540-60656-7. MR   0257325. Zbl   0176.00801.
• Griffiths, Phillip; Harris, Joseph (1978). Principles of algebraic geometry. Pure and Applied Mathematics. New York: John Wiley & Sons. ISBN   0-471-32792-1. MR   0507725. Zbl   0408.14001.
• Harvey, Reese; Lawson, H. Blaine Jr. (1982). "Calibrated geometries". Acta Mathematica . 148: 47–157. doi: 10.1007/BF02392726 . MR   0666108. Zbl   0584.53021.
• 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. MR   3674984. Zbl   1380.53003.
• 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. MR   1550581. Zbl   0015.07602.