Nonlinear Equations - UFRJ

[SEC. 5.5: COMPACTIFICATIONS 69
As a result, the Weyl inner product is invariant under unitary
action f f ◦ U ∗ and moreover,
K(Ux, Uy) = K(x, y).
Hence ω is ‘equivariant’ by U(n + 1). This action therefore generates
an action in quotient space P n . Moreover, U(n + 1) acts transitively
on P n , meaning that for all x, y ∈ P n there is U ∈ U(n + 1)
with y = Ux.
In this sense, P n is said to be ‘homogeneous’. The formal definition
states that a homogeneous manifold is a manifold that is quotient
of two Lie groups, and P n = U(n + 1)/(U(1) × U(n)).
We can now mimic the argument given for Theorem 1.3
Theorem 5.21. Let F 1 , . . . , F n be fewspaces of equations on M/H.
Suppose that
1. M/H is compact.
2. A group G acts transitively on M/H, in such a way that the
induced forms ω i on M/H are G-equivariant.
3. Assume furthermore that the set of regular values of π 1 : V → F
is path-connected.
Let f = f 1 , . . . , f n ∈ F = F 1 × · · · × F n . Then,
with equality almost everywhere.
n M/H (f) ≤ 1
π
∫M/H
n ω 1 ∧ · · · ∧ ω n
Proof. Let Σ be the set of critical values of F. From Sard’s Theorem
it has zero measure.
For all f, g ∈ F \ Σ, we claim that n M (f) ≥ n M (g). Indeed, there
is a path (f t ) t∈[0,1] in F \ Σ. By the inverse function theorem and
because M/H is compact, each root of f can be continued to a root
of g.
It follows that n M (f) is independent of f ∈ F \ Σ. Thus with
probability one,
n M (f) = 1
π
∫M
n ω 1 ∧ · · · ∧ ω n .