dωi = 2ρ § ωi ’ ±i § ωj + πi § θ,

j

it is tempting to de¬ne

Hi

πi = πi +

˜ nω ,

and rewrite them as

θ 0 πj

˜ θ

=’ §

d .

±i

ωi ωj

’˜i

π j

Observe that this looks exactly like the structure equation characterizing the

Levi-Civita connection of a Riemannian metric. We justify and use this as

follows.

Consider the quadratic form on B3

θ2 + (ωi )2 .

An easy computation shows that for any vertical vector ¬eld v ∈ Ker(π— ) for

π : B3 ’ Q,

Lv θ 2 + (ωi )2 = 0.

This means that our quadratic form is the pullback of a quadratic form on

Q, which de¬nes there a Riemannian metric ds2 . There is locally a bundle

isomorphism over Q

B3 ’ F(Q, ds2)

from B3 , which was constructed from the neo-classical Poincar´-Cartan form

e

Π, to the orthonormal frame bundle of this Riemannian metric. Under this iso-

morphism, the Q-semibasic forms θ, ω i correspond to the tautological semibasic

forms on F(Q, ds2), while the matrix

0 πj

˜

±i

’˜i

π j

corresponds to the Levi-Civita connection matrix. The contact manifold M , as a

quotient of B3 , may be then identi¬ed with the manifold of tangent hyperplanes

to Q; and the Poincar´-Cartan form is

e

’θ § (πi § ω(i) )

Π =

’θ § (˜i § ω(i) ’ Hω).

= π

´

78 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

We recognize this as exactly the Poincar´-Cartan form for the prescribed mean

e

curvature H = H(q) system, in an arbitrary (n + 1)-dimensional Riemannian

manifold. The following is what we have shown.

Theorem 2.5 A de¬nite neo-classical Poincar´-Cartan form (M, Π) whose dif-

e

ijk ij i

ferential invariants satisfy T = 0 and U = »δj with » < 0 is locally equiv-

alent to the Poincar´-Cartan of the prescribed mean curvature system on some

e

Riemannian manifold (Qn+1 , ds2).

We will consider these Poincar´-Cartan forms further in §4.1, when we discuss

e

the formula for the second variation of a Lagrangian functional FΛ. At that

time, we will also see an interpretation of the partial reduction B2 ⊃ B3 in

terms of the Riemannian geometry. Note that it is easy, given (M, Π) as in

the proposition, to determine the prescribed function H(q) by carrying out

the reductions described above, and to determine the Riemann curvature of

the ambient (n + 1)-manifold in terms of the connection 1-forms πi, ±i . The

˜ j

Euclidean minimal surface system discussed in §1.4 is the case H = 0, Rijkl = 0.

The fact that such an (M, Π) canonically determines (Q, ds2) implies the

following.8

Corollary 2.1 The symmetry group of (M, Π) is equal to the group of isome-

tries of (Q, ds2) that preserve the function H.

A consequence of this is the fact, claimed in §1.4, that all symmetries of the

minimal surface Poincar´-Cartan form”and hence, all classical conservation

e

laws for the Euler-Lagrange equation”are induced by Euclidean motions.

Finally, in case T ijk = 0 and U ij = »δj with » > 0 instead of » < 0, one

i

can carry out similar reductions, eventually producing on the quotient space

Qn+1 a Lorentz metric ds2 = ’θ2 + (ωi )2 ; the Poincar´-Cartan form is then

e

equivalent to that for prescribed mean curvature of space-like hypersurfaces.

8 As

usual, this assumes that the foliation associated to JΠ is simple; otherwise, only a

local reformulation holds.

Chapter 3

Conformally Invariant

Systems

Among non-linear Euler-Lagrange equations on Rn , the largest symmetry group

that seems to occur is the (n+1)(n+2) -dimensional conformal group. This consists

2

of di¬eomorphisms of the n-sphere that preserve its standard conformal struc-

ture, represented by the Euclidean metric under stereographic projection to Rn .

These maximally symmetric equations have a number of special properties, in-

cluding of course an abundance of classical conservation laws as predicted by

Noether™s theorem. This chapter concerns the geometry of the Poincar´-Cartan

e

forms associated to these equations, and that of the corresponding conservation

laws.

We will begin by presenting background material on conformal geometry.

This includes a discussion of the ¬‚at conformal structure on the n-sphere and

its symmetry group, a construction of a canonical parallelized principal bundle

over a manifold with conformal structure, and the de¬nition of the conformal

Laplacian, a second-order di¬erential operator associated to a conformal struc-

ture. This material will provide the framework for understanding the geometry

of non-linear Poisson equations, in particular the maximally symmetric non-

linear example

n+2

∆u = Cu n’2 , C = 0.

After developing the geometric context for this equation, we will continue the

equivalence problem for Poincar´-Cartan forms, pursuing the branch in which

e

these Euler-Lagrange equations occur.

We then turn to conservation laws for these conformally invariant equations.

The elaborate geometric structure allows several approaches to computing these

conservation laws, and we will carry out one of them in detail. The analogous

development for non-linear wave equations involves conformal structures with

Lorentz signature, and the conserved quantities for maximally symmetric Euler-

Lagrange equations in this case give rise to integral identities that have been

very useful in analysis.

79

80 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

3.1 Background Material on Conformal Geom-

etry

In this section, we discuss some of the less widely known aspects of conformal

geometry. In the ¬rst subsection, we de¬ne a ¬‚at model for conformal geome-

try which is characterized by its large symmetry group, and we give structure

equations in terms of the Maurer-Cartan form of this group. In the second sub-

section, we give Cartan™s solution to the local equivalence problem for general

conformal structures on manifolds. This consists of an algorithm by which one

associates to any conformal structure (N, [ds2 ]) a parallelized principal bundle

P ’ N having structure equations of a speci¬c algebraic form. In the third

subsection, we introduce a second-order di¬erential operator ∆, called the con-

formal Laplacian, which is associated to any conformal structure and which

appears in the Euler-Lagrange equations of conformal geometry that we study

in the remainder of the chapter. The fundamental de¬nition is the following.

De¬nition 3.1 A conformal inner-product at a point p ∈ N is an equiva-

lence class of positive inner-products on Tp N , where two such inner-products

are equivalent if one is a positive scalar multiple of the other. A conformal

structure, or conformal metric, on N consists of a conformal inner-product at

each point p ∈ N , varying smoothly in an obvious sense.

Note that this emphasizes the pointwise data of the conformal structure, unlike

the usual de¬nition of a conformal structure as an equivalence class of global

Riemannian metrics. An easy topological argument shows that these notions

are equivalent.

3.1.1 Flat Conformal Space

We start with oriented Lorentz space Ln+2 , with coordinates x = (x0 , . . . , xn+1),

orientation

dx0 § · · · § dxn+1 > 0,

and inner-product

x, y = ’(x0yn+1 + xn+1y0 ) + xi y i .

i

Throughout this section, we use the index ranges 0 ¤ a, b ¤ n + 1 and 1 ¤ i, j ¤

n.

A non-zero vector x ∈ Ln+2 is null if x, x = 0. A null vector x is positive

if x0 > 0 or xn+1 > 0; this designation is often called a “time-orientation”

for Ln+2 . The symmetries of Lorentz space are the linear transformations of

Ln+2 preserving the inner-product, the orientation, and the time-orientation,

and they constitute a connected Lie group SO o (n + 1, 1). We denote the space

of positive null vectors by

Q = {x ∈ Ln+2 : x, x = 0, and x0 > 0 or xn+1 > 0},

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 81

which is one half of the familiar light-cone, with axis {xi = x0 ’ xn+1 = 0}.

We now de¬ne ¬‚at conformal space R to be the space of null lines in Ln+2.

As a manifold, R is a non-singular quadric in the projective space P(Ln+2),

which is preserved by the natural action of the symmetry group SO o (n + 1, 1)

of Ln+2. We will describe the ¬‚at conformal structure on R below, in terms

of the Maurer-Cartan form of the group. Note that the obvious map Q ’ R,

which we will write as x ’ [x], gives a principal bundle with structure group

R— .

In the literature, R is usually de¬ned as Rn with a point added at in¬nity

to form a topological sphere. To make this identi¬cation, note that for x, y ∈ Q,

we have x, y ¤ 0, with equality if and only if [x] = [y]. We then claim that

def

Hy = {x ∈ Q : x, y = ’1}

is di¬eomorphic to both Rn and R\[y]; this is easily proved for y = (0, . . . , 0, 1),

for instance, where the map Rn ’ Hy is given by

(x1, . . . , xn) ’ (1, x1, . . . , xn, 1 ||x||2). (3.1)

2

The classical description of the conformal structure on R is obtained by trans-

porting the Euclidean metric on Rn to Hy , and noting that for y = y with

[y] = [y ], this gives unequal but conformally equivalent metrics on R\[y]. The

fact that SOo (n + 1, 1) acts transitively on R then implies that for [x] = [y] the

conformal structures obtained on R\[x] and R\[y] are the same.

A Lorentz frame is a positively oriented basis f = (e0 , . . . , en+1) of Ln+2, in

which e0 and en+1 positive null vectors, and for which the inner-product is (in

blocks of size 1, n, 1, like most matrices in this section)

«

0 ’1

0

e a , eb = 0 I n 0 .

’1 0 0

We let P denote the set of all Lorentz frames. There is a standard simply

transitive right-action of SO o (n + 1, 1) on P , by which we can identify the two

spaces in a way that depends on a choice of basepoint in P ; this gives P the

structure of a smooth manifold. Because we have used the right-action, the

pullback to P of any left-invariant 1-form on SO o (n + 1, 1) is independent of

this choice of basepoint. These pullbacks can be intrinsically described on P as

follows. We view each ea as a map P ’ Ln+2, and we de¬ne 1-forms ρ, ω i ,

βj , ±i on P by decomposing the Ln+2 -valued 1-forms dea in terms of the bases

j

{eb }: ±

de0 = 2e0 ρ + ei ωi ,

dej = e0 βj + ei ±i + en+1 ωj ,

j

den+1 = ei βi ’ 2en+1ρ.

Equivalently,

«

2ρ βj 0

ω i ±i βi .

d e0 ej en+1 = e0 ei en+1 j

0 ωj ’2ρ

82 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

These forms satisfy ±i + ±j = 0 but are otherwise linearly independent, and

j i

they span the left-invariant 1-forms on SO o (n + 1, 1) under the preceding iden-

ti¬cation with P . Decomposing the exterior derivatives of these equations gives

the Maurer-Cartan equations, expressed in matrix form as

« « «

2ρ βj 0 2ρ βj 0

2ρ βk 0

d ω i ±i βi + ω i ± i βi § ωk ±k βk = 0. (3.2)

j j

k

j k

0 ωj ’2ρ

0 ω ’2ρ ’2ρ

0ω

All of the local geometry of R that is invariant under SO o (n + 1, 1) can be

expressed in terms of these Maurer-Cartan forms. In particular, the ¬bers of

the map πR : P ’ R given by

πR : (e0 , . . . , en+1) ’ [e0]

are the integral manifolds of the integrable Pfa¬an system

IR = {ω1 , . . . , ωn }.

This ¬bration has the structure of a principal bundle, whose structure group

consists of matrices in SO o (n + 1, 1) of the form

«2