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 {{math|g}}, symplectic form {{math|ω}}, and almost-complex structure {{math|J}}, linked by {{math|ω(u, v) {{=}} g(J(u), v)}} for any vectors {{math|u}} and {{math|v}}. Then for any orthonormal vectors {{math|v₁, ..., 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 {{math|v₁, ..., v_2k}} is closed under the operation of {{math|J}}.{{sfnm|1a1=Federer|1y=1969|1loc=Section 1.8.2}}

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 {{math|ω ∧ ⋅⋅⋅ ∧ ω}} is equal to {{math|k!}}.{{sfnm|1a1=Federer|1y=1969|1loc=Section 1.8.2}}

Proof

={{math|''k'' {{=}} 1}}=

In the special case {{math|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 {{math|J(v₁)}} and {{math|v₂}} are collinear, which is equivalent to the span of {{math|v₁, v₂}} being closed under {{mvar|J}}.

={{math|''k'' > 1}}=

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

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

where {{math|ι}} denotes the inclusion map from {{mvar|T}} into {{mvar|V}}.{{sfnm|1a1=McDuff|1a2=Salamon|1y=2017|1loc=Lemma 2.4.5}} 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 {{math|k {{=}} 1}} case. If equality holds, then according to the {{math|k {{=}} 1}} equality case, it must be the case that {{math|ω(e_{2i − 1}, e_2i) {{=}} ±1}} for each {{mvar|i}}. This is equivalent to either {{math|ω(e_{2i − 1}, e_2i) {{=}} 1}} or {{math|ω(e_2i, e_{2i − 1}) {{=}} 1}}, which in either case (from the {{math|k {{=}} 1 }} case) implies that the span of {{math|e_{2i − 1}, e_2i}} is closed under {{math|J}}, and hence that the span of {{math|e₁, ..., e_2k}} is closed under {{mvar|J}}.

Finally, the dependence of the quantity

: $(\underbrace{\omega\wedge\cdots\wedge\omega}_{k\text{ times}})(v_1,\ldots,v_{2k})$

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

Consequences

Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any {{math|2k}}-dimensional embedded submanifold {{mvar|M}}, there is

: $\operatorname{vol}(M)\geq\frac{1}{k!}\int_M \omega^k,$

where {{math|ω}} is the Kähler form of the metric. Furthermore, equality is achieved if and only if {{mvar|M}} is a complex submanifold.{{sfnm|1a1=Griffiths|1a2=Harris|1y=1978|1loc=Section 0.2}} In the special case that the hermitian metric satisfies the Kähler condition, this says that {{math|{{sfrac|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 {{mvar|k}}.{{sfnm|1a1=Harvey|1a2=Lawson|1y=1982}} 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.{{sfnm|1a1=Federer|1y=1969|1loc=Section 5.4.19}}

Notes

References

{{cite book| last = Federer| first = Herbert| author-link1=Herbert Federer|title = Geometric measure theory| place= Berlin–Heidelberg–New York| publisher = Springer-Verlag| series = Die Grundlehren der mathematischen Wissenschaften| volume = 153| year = 1969| isbn = 978-3-540-60656-7| mr=0257325| zbl= 0176.00801 | doi=10.1007/978-3-642-62010-2}}
{{cite book|mr=0507725|last1=Griffiths|first1=Phillip|last2=Harris|first2=Joseph|title=Principles of algebraic geometry|series=Pure and Applied Mathematics|publisher=John Wiley & Sons|location=New York|year=1978|isbn=0-471-32792-1|zbl=0408.14001|author-link1=Phillip Griffiths|author-link2=Joe Harris (mathematician)}}
{{cite journal|mr=0666108|last1=Harvey|first1=Reese|last2=Lawson|first2=H. Blaine Jr.|title=Calibrated geometries|journal=Acta Mathematica|volume=148|year=1982|pages=47–157|doi=10.1007/BF02392726|zbl=0584.53021|author-link2=Blaine Lawson|author-link1=F. Reese Harvey|doi-access=free}}
{{cite book|mr=3674984|last1=McDuff|first1=Dusa|last2=Salamon|first2=Dietmar|title=Introduction to symplectic topology|edition=Third edition of 1995 original|series=Oxford Graduate Texts in Mathematics|publisher=Oxford University Press|location=Oxford|year=2017|isbn=978-0-19-879490-5|doi=10.1093/oso/9780198794899.001.0001|author-link1=Dusa McDuff|author-link2=Dietmar Salamon|zbl=1380.53003}}
{{cite journal|last1=Wirtinger|first1=W.|title=Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde in euklidischer und Hermitescher Maßbestimmung|year=1936|journal=Monatshefte für Mathematik und Physik|volume=44|pages=343–365|doi=10.1007/BF01699328|author-link1=Wilhelm Wirtinger|zbl=0015.07602|mr=1550581}}

Category:Inequalities (mathematics)

Category:Differential geometry

Category:Systolic geometry