3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM 107

We now begin to reduce B ’ M , as promised. To get information about

the derivative of the torsion coe¬cient A without knowing anything about dσ ij ,

we consider

d2(πi § ω(i) )

0=

n’2

d((n ’ 2)ρ § πi § ω(i) + § βi § ω(i) + Aθ § ω)

= 2θ

(dA + 4ρA) § θ § ω.

=

This describes the variation of the function A along the ¬bers of B ’ Q, where

we recall that Q is the leaf space of the integrable Pfa¬an system JΠ = {θ, ωi }.

In particular, we can write

dA + 4ρA = A0θ + Ai ωi , (3.46)

for some functions A0, Ai on B. We see that on each ¬ber of B ’ Q, either

A vanishes identically or A never vanishes, and we assume that the latter holds

throughout B. This is motivated by the case of the Poisson equation (3.33),

˜ 1

for which A = n fu (so we are assuming in particular that the zero-order term

f(x, u) depends on u). Because the sign of A is ¬xed, we assume A > 0 in what

follows. The case A < 0 is similar, but the case A = 0 is quite di¬erent.

For the ¬rst reduction of B ’ M , we de¬ne

1

B1 = {b ∈ B : A(b) = 4 } ‚ B.

From equation (3.46) with the assumption A > 0 everywhere, it is clear that

B1 ’ M is a principal subbundle of B, whose structure group™s Lie algebra

consists of matrices (3.31) with r = 0. Furthermore, restricted to B1 there is a

relation

ρ = A0 θ + Ai ω i . (3.47)

˜˜ ˜

In the case of a Poisson equation (3.33), our section (θ, ωi , πi) of B ’

J 1 (Rn, R) is generally not a section of B1 ‚ B, because we have along this

˜

section that A = fu . However, (3.46) guides us in ¬nding a section of B1 .

n

Namely, we de¬ne a function r(x, u, q) > 0 on M by

˜

r 4 = 4A = 4

n fu , (3.48)

and then one can verify that for the coframing

« « 2’n «

˜

ˆ r 0 0 θ

θ

ωi def 0 ωj ,

r 2 δj

i

0

= ˜

ˆ

j

’n

πi

ˆ πj

˜

0 0 r δi

one has the structure equation (3.42), with

ˆ ˆ1

ρ = r’1 dr = 1 fu dfu ,

’1

βi = ±i = 0,

ˆ ˆj A = 4.

4

108 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Again, we won™t have any need for σij . Observe that along this section of B1 ,

ˆ

ˆˆ ˆˆ

1 ’1 i

ρ = 4 fu dfu = A0 θ + Ai ω , so that

ˆ

ˆ ˆ

’1

Ai = 1 r’2 fu (fuxi + fuuqi ),

’1

A0 = 1 rn’2fu fuu, (3.49)

4 4

with r given by (3.48).

Returning to the general situation on B1 , we di¬erentiate (3.47) and ¬nd

(dA0 ’ (n ’ 2)ρA0 ) § θ + (dAi + 2ρAi ’ Aj ±j + 2 βi ’ A0πi ) § ωi = 0,

1

i

and the Cartan lemma then gives

dA0 ’ (n ’ 2)ρA0 = A00θ + A0iωi , (3.50)

dAi + 2ρAi ’ Aj ±j + 2 βi ’ A0 πi = Ai0θ + Aij ωj ,

1

(3.51)

i

with A0i = Ai0 and Aij = Aji.

We interpret (3.50) as saying that if A0 vanishes at one point of a ¬ber of

B1 ’ Q, then it vanishes everywhere on that ¬ber. We make the assumption

that A0 = 0; the other extreme case, where A0 = 0 everywhere, gives a di¬erent

branch of the equivalence problem. Note that in the case of a Poisson equation

(3.33), the condition A0 = 0 implies by (3.49) that the equation is everywhere

non-linear. This justi¬es our decision to pursue, among the many branches of

the equivalence problem within the larger conformal branch, the case A > 0,

A0 = 0. This justi¬cation was our main reason to carry along the example of

the Poisson equation, and we will not mention it again. General calculations

involving it become rather messy at this stage, but how to continue should be

clear from the preceding.

Returning to the general setting, our second reduction uses (3.51), which

tells us that the locus

B2 = {b ∈ B1 : Ai (b) = 0} ‚ B1

is a principal subbundle of B1 ’ M , whose structure group™s Lie algebra consists

of matrices (3.31) with r = di = 0. Furthermore, restricted to B2 there are

relations

βi = 2(Ai0θ + Aij ωj + A0 πi),

and also

ρ = A0 θ.

With the Ai out of the way we di¬erentiate once more, and applying the

Cartan lemma ¬nd that on B2 , modulo {θ, ωi , πi},

≡

dA00 0, (3.52)

A0j ±j ,

≡

dA0i (3.53)

i

Akj ±k + Aik ±k + A0 σij .

≡

dAij (3.54)

i j

3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM 109

We interpret (3.52) as saying that A00 descends to a well-de¬ned function on

M . We interpret (3.53) as saying that the vector valued function (A0i) repre-

sents a section a vector bundle associated to B2 ’ M . We interpret (3.54) as

saying that if A0 = 0, then the matrix (Aij ) represents a section of a vector

bundle associated to B2 ’ M . However, we have already made the assump-

tion that A0 = 0 everywhere. In some examples of interest, most notably for

n+2

the equation ∆u = Cu n’2 , the section (A0i) vanishes; for a general non-linear

Poisson equation, this vanishing loosely corresponds to the non-linearity being

translation-invariant on ¬‚at conformal space. We will not need to make any

assumptions about this quantity.

This allows us to make a third reduction. With A0 = 0, (3.54) tells us that

the locus where the trace-free part of Aij vanishes,

def

B3 = {b ∈ B2 : A0 (b) = Aij (b) ’ n δij Akk(b) = 0},

1

ij

is a subbundle B3 ’ M of B2 ’ M . In terms of (3.31), the Lie algebra of the

structure group of B3 is de¬ned by r = di = sij = 0.

Let us summarize what we have done. Starting from the structure equations

(3.34, 3.35, 3.36, 3.37) on B ’ M for a de¬nite, neo-classical Poincar´-Cartan

e

form with n ≥ 3, we specialized to the case where the torsion satis¬es

U ij = Sj = T ijk = 0.

i

In this case, we found that the leaf space N of the Pfa¬an system {ω 1 , . . . , ωn }

(ωi )2 . We replaced each pseudo-

has a conformal structure pulling back to

connection form δi by its multiple βi , and guided by computations from con-

formal geometry, we determined the torsion in the equation for d±i , which re-

j

sembled a Riemann curvature tensor. This torsion™s analog of scalar curvature

provided our fundamental invariant A, which had ¬rst “covariant derivatives”

A0 , Ai , and second “covariant derivatives” A00, Ai0 = A0i, Aij = Aji. With

the assumptions

A = 0, A0 = 0,

we were able to make successive reductions by passing to the loci where

1 1

A = 4, Ai = 0, Aij = n δij Akk .

This leaves us on a bundle B3 ’ M with a coframing ω i , ρ, βi, ±i , satis-

j

fying structure equations exactly like those on the conformal bundle P ’ N

associated with (N, [ds2]). From here, a standard result shows that there is a

local di¬eomorphism B3 ’ P under which the two coframings correspond. In

particular, the invariants Ai and Bjkl remaining in the bundle B3 equal the

jkl

invariants named similarly in the conformal structure, so we can tell for example

if the conformal structure associated to our Poincar´-Cartan form is ¬‚at.

e

We now write the restricted Poincar´-Cartan form,

e

’θ § (πi § ω(i) )

Π=

’ n Akk ωi § ω(i)

1 1 1 1

’ A0 ρ §

= β

2i

A0

1

’ 2A2 ρ § (βi § ω(i) ’ 2Akkω).

=

0

110 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

We can see from previous equations that Akk is constant on ¬bers of B3 ’

M . Therefore, it makes sense to say that Akk is or is not constant on B3 . If

it is constant, and if the conformal structure on N is ¬‚at (that is, Ai = 0 if

jkl

n ≥ 4, or Bjkl = 0 if n = 3), then our Poincar´-Cartan form is equivalent to

e

that associated to the non-linear Poisson equation

n+2

∆u = Cu n’2 ,

where C = (n ’ 2)Akk . This completes the characterization of Poincar´-Cartan

e

forms equivalent to that of this equation. Our next goal is to determine the

conservation laws associated to this Poincar´-Cartan form.

e

n+2

3.4 Conservation Laws for ∆u = Cu n’2

In this section, we will determine the classical conservation laws for the confor-

mally invariant non-linear Poisson equation

n+2

∆u = Cu n’2 . (3.55)

Recall that from Λ0 = ω and Λ1 = βi § ω(i) we constructed the functional

1 C

’

Λ= 2(n’2) Λ1 n(n’2) Λ0

having the Poincare-Cartan form

2C

Π = dΛ = ρ § (βi § ω(i) ’ ω),

n’2

and that under a certain embedding σ : J 1(Rn , R+) ’ P , the Euler-Lagrange

system of Π restricted to a transverse Legendre submanifold is generated by

n+2

Ψ = ∆u ’ Cu n’2 dx,

for coordinates on J 1 (Rn, R+ ) described in the proof of Proposition 3.2. We

also proved that the composition of σ : J 1 (Rn, R+ ) ’ P with the projection

P ’ M gives an open contact embedding of J 1 (Rn, R+ ) as a dense subset of

M . Our invariant forms on P pull back via σ to give the following forms on

J 1 (Rn , R+), expressed in terms of the canonical coordinates (xi, u, pi):

± ’1 i

1 1

ρ = n’2 u du ’ 2 pi dx ,

i

ω = u n’2 dxi,

2

(3.56)

2

2

βi = u’ n’2 dpi ’ pi pj dxj + ||p|| dxi ,