- Source: 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, ..., v2k there is
(
ω
∧
⋯
∧
ω
⏟
k
times
)
(
v
1
,
…
,
v
2
k
)
≤
k
!
.
{\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, ..., v2k is closed under the operation of J.
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!.
Proof
= k = 1
=In the special case k = 1, the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality:
ω
(
v
1
,
v
2
)
=
g
(
J
(
v
1
)
,
v
2
)
≤
‖
J
(
v
1
)
‖
g
‖
v
2
‖
g
=
1.
{\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, ..., v2k be fixed, and let T denote their span. Then there is an orthonormal basis e1, ..., e2k of T with dual basis w1, ..., w2k such that
ι
∗
ω
=
∑
j
=
1
k
ω
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e
2
j
−
1
,
e
2
j
)
w
2
j
−
1
∧
w
2
j
,
{\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. This implies
ι
∗
ω
∧
⋯
∧
ι
∗
ω
⏟
k
times
=
k
!
∏
i
=
1
k
ω
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e
2
i
−
1
,
e
2
i
)
w
1
∧
⋯
∧
w
2
k
,
{\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
(
ω
∧
⋯
∧
ω
⏟
k
times
)
(
e
1
,
…
,
e
2
k
)
=
k
!
∏
i
=
1
k
ω
(
e
2
i
−
1
,
e
2
i
)
≤
k
!
,
{\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(e2i − 1, e2i) = ±1 for each i. This is equivalent to either ω(e2i − 1, e2i) = 1 or ω(e2i, e2i − 1) = 1, which in either case (from the k = 1 case) implies that the span of e2i − 1, e2i is closed under J, and hence that the span of e1, ..., e2k is closed under J.
Finally, the dependence of the quantity
(
ω
∧
⋯
∧
ω
⏟
k
times
)
(
v
1
,
…
,
v
2
k
)
{\displaystyle (\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})}
on v1, ..., v2k is only on the quantity v1 ∧ ⋅⋅⋅ ∧ v2k, and from the orthonormality condition on v1, ..., v2k, this wedge product is well-determined up to a sign. This relates the above work with e1, ..., e2k to the desired statement in terms of v1, ..., v2k.
Consequences
Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any 2k-dimensional embedded submanifold M, there is
vol
(
M
)
≥
1
k
!
∫
M
ω
k
,
{\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. In the special case that the hermitian metric satisfies the Kähler condition, this says that 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. 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.
See also
Gromov's inequality for complex projective space
Systolic geometry
Notes
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.
Kata Kunci Pencarian:
- Wirtinger inequality (2-forms)
- Wirtinger's inequality
- Wirtinger's inequality for functions
- Poincaré inequality
- Wilhelm Wirtinger
- List of inequalities
- List of differential geometry topics
- Symplectic manifold
- Differential form
- Gromov's inequality for complex projective space
