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dОё = в€’ПЂi в€§ П‰i ,
dП‰i = 2ПЃ в€§ П‰i в€’ О±i в€§ П‰j + ПЂi в€§ Оё,
j

it is tempting to deп¬Ѓne
Hi
ПЂi = ПЂi +
Лњ nП‰ ,
and rewrite them as
Оё 0 ПЂj
Лњ Оё
=в€’ в€§
d .
О±i
П‰i П‰j
в€’Лњi
ПЂ 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

Оё2 + (П‰i )2 .

An easy computation shows that for any vertical vector п¬Ѓeld v в€€ Ker(ПЂв€— ) for
ПЂ : B3 в†’ Q,
Lv Оё 2 + (П‰i )2 = 0.
This means that our quadratic form is the pullback of a quadratic form on
Q, which deп¬Ѓnes there a Riemannian metric ds2 . There is locally a bundle
isomorphism over Q
B3 в†’ F(Q, ds2)
from B3 , which was constructed from the neo-classical PoincarВґ-Cartan form
e
О , to the orthonormal frame bundle of this Riemannian metric. Under this iso-
morphism, the Q-semibasic forms Оё, П‰ i correspond to the tautological semibasic
forms on F(Q, ds2), while the matrix
0 ПЂj
Лњ
О±i
в€’Лњi
ПЂ j

corresponds to the Levi-Civita connection matrix. The contact manifold M , as a
quotient of B3 , may be then identiп¬Ѓed with the manifold of tangent hyperplanes
to Q; and the PoincarВґ-Cartan form is
e
в€’Оё в€§ (ПЂi в€§ П‰(i) )
О  =
в€’Оё в€§ (Лњi в€§ П‰(i) в€’ HП‰).
= ПЂ
Вґ
78 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

We recognize this as exactly the PoincarВґ-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 deп¬Ѓnite neo-classical PoincarВґ-Cartan form (M, О ) whose dif-
e
ijk ij i
ferential invariants satisfy T = 0 and U = О»Оґj with О» < 0 is locally equiv-
alent to the PoincarВґ-Cartan of the prescribed mean curvature system on some
e
Riemannian manifold (Qn+1 , ds2).

We will consider these PoincarВґ-Cartan forms further in В§4.1, when we discuss
e
the formula for the second variation of a Lagrangian functional FО›. At that
time, we will also see an interpretation of the partial reduction B2 вЉѓ B3 in
terms of the Riemannian geometry. Note that it is easy, given (M, О ) 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 ПЂi, О±i . The
Лњ j
Euclidean minimal surface system discussed in В§1.4 is the case H = 0, Rijkl = 0.

The fact that such an (M, О ) canonically determines (Q, ds2) implies the
following.8

Corollary 2.1 The symmetry group of (M, О ) 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 В§1.4, that all symmetries of the
minimal surface PoincarВґ-Cartan formвЂ”and hence, all classical conservation
e
laws for the Euler-Lagrange equationвЂ”are induced by Euclidean motions.
Finally, in case T ijk = 0 and U ij = О»Оґj with О» > 0 instead of О» < 0, one
i

can carry out similar reductions, eventually producing on the quotient space
Qn+1 a Lorentz metric ds2 = в€’Оё2 + (П‰i )2 ; the PoincarВґ-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 JО  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 diп¬Ђeomorphisms 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
NoetherвЂ™s theorem. This chapter concerns the geometry of the PoincarВґ-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 п¬‚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 deп¬Ѓnition of the conformal
Laplacian, a second-order diп¬Ђerential 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
в€†u = Cu nв€’2 , C = 0.
After developing the geometric context for this equation, we will continue the
equivalence problem for PoincarВґ-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 п¬Ѓrst subsection, we deп¬Ѓne a п¬‚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 CartanвЂ™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 в†’ N having structure equations of a speciп¬Ѓc algebraic form. In the third
subsection, we introduce a second-order diп¬Ђerential operator в€†, 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 deп¬Ѓnition is the following.
Deп¬Ѓnition 3.1 A conformal inner-product at a point p в€€ 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 в€€ N , varying smoothly in an obvious sense.
Note that this emphasizes the pointwise data of the conformal structure, unlike
the usual deп¬Ѓnition 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 в€§ В· В· В· в€§ dxn+1 > 0,
and inner-product

x, y = в€’(x0yn+1 + xn+1y0 ) + xi y i .
i

Throughout this section, we use the index ranges 0 в‰¤ a, b в‰¤ n + 1 and 1 в‰¤ i, j в‰¤
n.
A non-zero vector x в€€ 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 вЂњtime-orientationвЂќ
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 в€€ 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 в€’ xn+1 = 0}.
We now deп¬Ѓne п¬‚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 п¬‚at conformal structure on R below, in terms
of the Maurer-Cartan form of the group. Note that the obvious map Q в†’ R,
which we will write as x в†’ [x], gives a principal bundle with structure group
Rв€— .
In the literature, R is usually deп¬Ѓned as Rn with a point added at inп¬Ѓnity
to form a topological sphere. To make this identiп¬Ѓcation, note that for x, y в€€ Q,
we have x, y в‰¤ 0, with equality if and only if [x] = [y]. We then claim that
def
Hy = {x в€€ Q : x, y = в€’1}
is diп¬Ђeomorphic to both Rn and R\[y]; this is easily proved for y = (0, . . . , 0, 1),
for instance, where the map Rn в†’ Hy is given by
(x1, . . . , xn) в†’ (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)
пЈ« пЈ¶
0 в€’1
0
e a , eb = пЈ­ 0 I n 0 пЈё .
в€’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 в†’ Ln+2, and we deп¬Ѓne 1-forms ПЃ, П‰ i ,
ОІj , О±i on P by decomposing the Ln+2 -valued 1-forms dea in terms of the bases
j
{eb }: пЈ±
пЈІ de0 = 2e0 ПЃ + ei П‰i ,
dej = e0 ОІj + ei О±i + en+1 П‰j ,
j
пЈі
den+1 = ei ОІi в€’ 2en+1ПЃ.
Equivalently,
пЈ« пЈ¶
2ПЃ ОІj 0
пЈ­ П‰ i О±i ОІi пЈё .
d e0 ej en+1 = e0 ei en+1 j
0 П‰j в€’2ПЃ
82 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

These forms satisfy О±i + О±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-
tiп¬Ѓcation with P . Decomposing the exterior derivatives of these equations gives
the Maurer-Cartan equations, expressed in matrix form as
пЈ« пЈ¶пЈ« пЈ¶пЈ« пЈ¶
2ПЃ ОІj 0 2ПЃ ОІj 0
2ПЃ ОІk 0
d пЈ­ П‰ i О±i ОІi пЈё + пЈ­ П‰ i О± i ОІi пЈё в€§ пЈ­ П‰k О±k ОІk пЈё = 0. (3.2)
j j
k
j k
0 П‰j в€’2ПЃ
0 П‰ в€’2ПЃ в€’2ПЃ
0П‰

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 п¬Ѓbers of
the map ПЂR : P в†’ R given by

ПЂR : (e0 , . . . , en+1) в†’ [e0]

are the integral manifolds of the integrable Pfaп¬ѓan system

IR = {П‰1 , . . . , П‰n }.

This п¬Ѓbration has the structure of a principal bundle, whose structure group
consists of matrices in SO o (n + 1, 1) of the form
пЈ«2 пЈ¶
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