1

dρ =

We therefore have a situation similar to that in §3.3 (cf. (3.40, 3.41)), with

equations formally like those in the conformal geometry equivalence problem of

§3.1.2. We can mimic the derivation of conformal structure equations in the

2

present case by ¬rst setting βi = n’2 δi , and then we know that this results in

an equation

d±i + ±i § ±k + βi § ωj ’ βj § ωi = 2 Ai ωk § ωl ,

1

(4.6)

j k j jkl

where the quantity (Ai ) has the symmetries of the Riemann curvature tensor.

jkl

Furthermore, we know that there are unique functions tij = tji such that replac-

ing βi ; βi + tij ωj will yield the preceding equation with Al = 0. However,

jkl

it will simplify matters later if we go back and replace instead

n’2 o j

δi ; δ i + 2 tij ω ,

1

where to = tij ’ n δij tkk is the traceless part (note that only a traceless addition

ij

to δi will preserve the structure equations on B ’ M ). In terms of the new δi ,

we de¬ne

def

’ n Kωi ),

2 1

βi = (δ (4.7)

n’2 i

138 CHAPTER 4. ADDITIONAL TOPICS

where

def

K = ’ n’2 tjj .

2

This K was chosen so that de¬ning βi by (4.7) gives the correct conformal

structure equation (4.6) with Al = 0; it re¬‚ects the di¬erence between the

jkl

pseudo-connection forms for the Poincar´-Cartan form and the Cartan con-

e

nection forms for the induced conformal geometry on the submanifold. One

interpretation is the following.

The function K on Bf is a fundamental invariant of a stationary

submanifold (N, [ds2]f ) ’ (M, Π) of an Euler-Lagrange system, and

may be thought of as an extrinsic curvature depending on up to third

derivatives of the immersion f.

In the classical setting, N is already the 1-jet graph of a solution z(x) of an Euler-

Lagrange equation, so an expression for K(x) depends on fourth derivatives of

z(x).

Now suppose that our integral manifold f : N ’ M , with δi , βi , and K as

above, is the initial manifold F0 = f in a Legendre variation F : N —[0, 1] ’ M .

Then we can rewrite our formula (4.5) for the second variation as

g(dgi + nρgi ’ gj ±j +

δ 2 (FΛ )N0 (g) = ’ § ω(i) + g2 Kω .

n’2

2 gβi )

i

N0

Part of this integrand closely resembles the expression (3.19) for the second co-

variant derivative of a section of a density line bundle, discussed in constructing

the conformal Laplacian in §3.1.3. This suggests the following computations.

First, consider the structure equation dθ = ’(n ’ 2)ρ § θ ’ πi § ωi . Along

W0 (but not yet restricted to W0 ), where θ = g dt and πi = gidt, this reads

dg § dt = ’(n ’ 2)ρ § g dt ’ gidt § ωi ,

so that restricted to W0 , we have

dg + (n ’ 2)ρg = giωi .

This equation should be interpreted on Bf , which is identi¬ed with the principal

bundle associated to the conformal structure [ds2]f . It says that g is a section

n’2

of the density bundle D 2n , and that gi are the components of its covariant

derivative (see (3.18) ¬.). We can now write

dgi + nρgi ’ gj ±j + = gij ωj ,

n’2

gβi

i 2

and by de¬nition

n+2

∆f g = gii ∈ “(D ),

2n

where ∆f is the conformal Laplacian on N induced by f = F0 : N ’ M . We

now have a more promising version of the second variation formula:

δ 2 (FΛ )N0 (g) = ’ (g∆f g + g2 K)ω. (4.8)

N0

4.1. THE SECOND VARIATION 139

It is worth noting that the sign of this integrand does not depend on the sign

of the variation™s generating function g, and if we ¬x an orientation of N , then

the integrand Kω on Bf has a well-de¬ned sign at each point of N .

4.1.3 Intrinsic Integration by Parts

In order to detect local minima using the second variation formula (4.8), it is

often helpful to convert an integral like g∆g dx into one like ’ || g||2dx. In

the Euclidean setting, with either compact supports or with boundary terms,

this is done with integration by parts and is straightforward; the two integrands

di¬er by the divergence of g g, whose integral depends only on boundary data.

We would like to perform a calculation like this on N , for an arbitrary

Legendre variation g, using only intrinsic data. In other words, we would like

to associate to any N and g some ξ ∈ „¦n’1(N ) such that dξ is the di¬erence

between (g∆f g)ω and some quadratic expression Q( g, g)ω, possibly with

some additional zero-order terms. A natural expression to consider, motivated

by the ¬‚at case, is

ξ = ggi ω(i) . (4.9)

Here, in order for gi to make sense, we are assuming that we have a local

coframing on M adapted to the integral manifold N , so that at points of N ,

θ = g dt, πi = gi dt, and so that restricted to N , θ|N = πi |N = 0. We can then

compute:

= dg § gi ω(i) + g dgi § ω(i) + ggi dωj § ω(ij)

dξ (4.10)

= (dg + (n ’ 2)ρg) § giω(i) + g(dgi + nρgi ’ gj ±j ) § ω(i) (4.11)

i

(gi )2 + g∆f g ω ’ n’2 2

§ ω(i) .

= 2 g βi (4.12)

Now, the ¬rst term is exactly what we are looking for, and second is fairly

harmless because it is of order zero in the variation g, and in practice contributes

only terms similar to g 2 Kω.

The problem is that ξ (4.9) is de¬ned on the total space Bf ’ N , and

although semibasic for this bundle, it is not basic; that is, there is no form on

N that pulls back to Bf to equal ξ, even locally. The criterion for ξ to be basic

is that dξ be semibasic, and this fails because of the appearance of βi in (4.12).

But suppose that we can ¬nd some canonical reduction of the ambient B ’

M to a subbundle on which δi becomes semibasic over M ; in terms of the Lie

algebra (4.1), this means that we can reduce to the subgroup having Lie algebra

given by {di = 0}. In this case, each βi = n’2 (δi ’ K ωi ) is a linear combination

2

n

of θ, ωi , πi , and is in particular semibasic over N . Consequently, ξ is basic over

N , and we can perform the integration by parts in an invariant manner.

Unfortunately, there are cases in which no such canonical reduction of B is

possible. An example is the homogeneous Laplace equation on Rn ,

∆z = 0,

140 CHAPTER 4. ADDITIONAL TOPICS

which is preserved under an action of the conformal group SO o (n + 1, 1). The

associated conformal geometry on the trivial solution z = 0 is ¬‚at, and our

second variation formula reads

δ 2 (FΛ )0 (g) = g∆g dx

„¦

∞

for a variation g ∈ C0 („¦), „¦ ‚ Rn . It follows from our construction that this

integrand is invariant under a suitable action of the conformal group. However,

the tempting integration by parts

|| g||2dx

g∆g dx = ’

„¦ „¦

leaves us with an integrand which is not conformally invariant. It is this phe-

nomenon that we would like to avoid.

To get a sense of when one might be able to ¬nd a canonical subbundle of

B ’ M on which the δ i are semibasic, recall that

(n ’ 2)dρ ≡ ’δi § ωi (mod {θ, πi}).

Working modulo {θ, πi} essentially amounts to restricting to integral manifolds

of the Euler-Lagrange system. The preceding then says that a choice of sub-

bundle B ‚ B on which δi are semibasic gives a subbundle of each conformal

bundle Bf ’ N on which dρ is semibasic over N . Now, typically the role of ρ in

the Cartan connection for a conformal structure is as a psuedo-connection in the

density line bundle D; a special reduction is required for ρ to be a genuine con-

nection, and the latter requirement is equivalent to dρ being semibasic. In other

words, being able to integrate by parts in an invariant manner as described above

is equivalent to having a connection in D represented by the pseudo-connection

ρ. One way to ¬nd a connection in D is to suppose that D has somehow been

trivialized, and this is equivalent to choosing a Riemannian metric representing

the conformal class. This suggests that Euler-Lagrange systems whose integral

manifolds have canonical Riemannian metrics will have canonical reductions of

this type.

In fact, we have seen an example of a Poincar´-Cartan form whose geometry

e

B ’ M displays this behavior. This is the system for Riemannian hypersurfaces

having prescribed mean curvature, characterized in §2.5 in terms of di¬erential

invariants of its neo-classical, de¬nite Poincar´-Cartan form. To illustrate the

e

preceding discussion, we calculate the second variation formula for this system.

The reader should note especially how use of the geometry of the Poincar´- e

Cartan form gives a somewhat simpler derivation of the formula than one ¬nds

in standard sources.1

1 See for example pp. 513“539 of [Spi75], where the calculation is prefaced by a colorful

warning about its di¬culty.

4.1. THE SECOND VARIATION 141

4.1.4 Prescribed Mean Curvature, Revisited

In §2.5, we considered a de¬nite, neo-classical Poincar´-Cartan form (M, Π)

e

whose associated geometry (B ’ M, •) had invariants satisfying

T ijk = 0, U ij = »δ ij .

We further assumed the open condition

» < 0,

and this led to a series of reductions of B ’ M , resulting in a principal sub-

bundle B3 ’ M , having structure group with Lie algebra

±«

00 0

0 : ai + a j = 0 ,

0 ai

g3 = j j i

j

0 0 ’ai

on which the original structure equations (4.2, 4.3, 4.4) hold, with U ij = ’δ ij ,

Sj = T ijk = 0, ρ = ’ 2n θ for a function H on B3 , and dρ = ’ n’2 δi § ωj , where

H 1

i

δi ≡ 0 (mod {θ, ωj , πj }). In this case we computed that dH ≡ 0 (mod {θ, ω i }),

so that H de¬nes a function on the local quotient space Qn+1, which also inherits

a Riemannian metric (ωi )2 . The contact manifold M can be locally identi¬ed

with the bundle of tangent hyperplanes of Q, and the integral manifolds of the

Euler-Lagrange di¬erential system EΛ are the tangent loci of hypersurfaces in Q

whose mean curvature coincides with the background function H. In this case

B3 ’ M ’ Q is locally identi¬ed with the orthonormal frame bundle of the

Riemannian manifold Q.

An important point here is that the Riemannian geometry associated to Π

only appears after reducing to B3 ’ M . However, if our goal is to see the

formula for the second variation, then we face the following di¬culty. That

formula required the use of coframes of M adapted to a stationary submanifold

N ’ M in a certain way, but while adapted coframes can always be found

in B ’ M , there is no guarantee that they can be found in the subbundle

B3 ’ M , where the Riemannian geometry is visible.

We will overcome these di¬culties and illustrate the invariant calculation of

the second variation by starting only with the Riemannian geometry of (Q, ds 2).

This is expressed in the Levi-Civita connection in the orthonormal frame bundle,

where we can also give the Poincar´-Cartan form and Euler-Lagrange system

e

for prescribed mean curvature. We then introduce higher-order data on a larger

bundle, which allows us to study the second fundamental form. In fact, this

larger bundle corresponds to the partial reduction B2 ’ M on which ρ and δi

are semibasic, but σij is not. The end result of our calculation is formula (4.16).

In the following discussion, index ranges are 0 ¤ a, b, c ¤ n and 1 ¤ i, j, k ¤ n.

We begin with a generalization of the discussion in §1.4 of constant mean

curvature hypersurfaces in Euclidean space. Let (Q, ds2) be an oriented Rieman-

nian manifold of dimension n + 1. A frame for Q is a pair f = (q, e) consisting

142 CHAPTER 4. ADDITIONAL TOPICS

of a point q ∈ Q and a positively-oriented orthonormal basis e = (e0 , . . . , en)

for Tq Q. The set F of all such frames is a manifold, and the right SO(n + 1, R)-

action

a a a

(q, (e0 , . . . , en )) · (gb ) = (q, ( ea g0 , . . . , ea gn ))

gives the basepoint map

q:F ’Q