Wirtinger inequality (2-forms)

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.

For other inequalities named after Wirtinger, see Wirtinger's inequality.

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 $v 1, ..., v 2 k$ 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 $v 1, ..., v 2 k$ 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 $ω \land \cdot\cdot\cdot \land ω$ 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:

\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 $v 1, ..., v 2 k$ be fixed, and let $T$ denote their span. Then there is an orthonormal basis $e 1, ..., e 2 k$ of $T$ with dual basis $w 1, ..., w 2 k$ 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 $V$ .[2] 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 $ω (e 2 i - 1, e 2 i) = \pm1$ for each $i$ . This is equivalent to either $ω (e 2 i - 1, e 2 i) = 1$ or $ω (e 2 i, e 2 i - 1) = 1$ , which in either case (from the $k = 1$ case) implies that the span of $e 2 i - 1, e 2 i$ is closed under $J$ , and hence that the span of $e 1, ..., e 2 k$ 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 $v 1, ..., v 2 k$ is only on the quantity $v 1 \land \cdot\cdot\cdot \land v 2 k$ , and from the orthonormality condition on $v 1, ..., v 2 k$ , this wedge product is well-determined up to a sign. This relates the above work with $e 1, ..., e 2 k$ to the desired statement in terms of $v 1, ..., v 2 k$ .

Consequences

Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any $2 k$ -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.[3] In the special case that the hermitian metric satisfies the Kähler condition, this says that $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/k!ωk$ 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]

Notes

Federer 1969, Section 1.8.2.
McDuff & Salamon 2017, Lemma 2.4.5.
Griffiths & Harris 1978, Section 0.2.
Harvey & Lawson 1982.
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.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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.

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[FOOTNOTEFederer1969Section_1.8.2-1] Federer 1969, Section 1.8.2.

[FOOTNOTEMcDuffSalamon2017Lemma_2.4.5-2] McDuff & Salamon 2017, Lemma 2.4.5.

[FOOTNOTEGriffithsHarris1978Section_0.2-3] Griffiths & Harris 1978, Section 0.2.

[FOOTNOTEHarveyLawson1982-4] Harvey & Lawson 1982.

[FOOTNOTEFederer1969Section_5.4.19-5] Federer 1969, Section 5.4.19.