M 2n+1 = {(q, e0) : q ∈ Q, e0 ∈ Tq Q, ||e0|| = 1}

is identi¬ed with the Grassmannian bundle of oriented tangent n-planes in T Q,

and it has a contact structure generated by the 1-form

θ(q,e0 ) (v) = ds2 (e0 , q—(v)), v ∈ T(q,e0 ) M, (4.13)

where q : M ’ Q is the projection. An immersed oriented hypersurface in Q

has a unit normal vector ¬eld, which may be thought of as a 1-jet lift of the

submanifold to M . The submanifold of M thus obtained is easily seen to be a

Legendre submanifold for this contact structure, and the transverse Legendre

submanifold is locally of this form.

To carry out calculations on M , and even to verify the non-degeneracy of

θ, we will use the projection F ’ M de¬ned by (q, (e0, . . . , en)) ’ (q, e0 ).

Calculations can then be carried out using structure equations for the canon-

ical parallelization of F, which we now introduce. First, there are the n + 1

tautological 1-forms

•a 2 1

(q,e) = ds (ea , q— (·)) ∈ „¦ (F),

which form a basis for the semibasic 1-forms over Q. Next, there are globally

de¬ned, uniquely determined Levi-Civita connection forms •a = ’•b ∈ „¦1(F)

a

b

satisfying

d•a = ’•a § •b ,

b

(4.14)

d•a = ’•a § •c + 1 Ra •c § •d .

c 2 bcd

b b

The functions Ra on F are the components of the Riemann curvature tensor

bcd

with respect to di¬erent orthonormal frames, and satisfy

Ra + Ra = Ra + Rb = Ra + Ra + Ra = 0.

bcd bdc bcd acd bcd cdb dbc

We now distinguish the 1-form

def

θ = •0 ∈ „¦1(F),

which is the pullback via F ’ M of the contact 1-form given the same name in

(4.13). One of our structure equations now reads

dθ = ’•0 § •i . (4.15)

i

4.1. THE SECOND VARIATION 143

This implies that the original θ ∈ „¦1(M ) is actually a contact form, and also

that (θ, •i , •0 ) is a basis for the semibasic 1-forms for F ’ M .

i

At this point, we can give the Poincar´-Cartan form for the prescribed mean

e

∞

curvature system. Namely, let H ∈ C (Q) be a smooth function, and de¬ne

on F the (n + 1)-form

def

Π = ’θ § (•0 § •(i) ’ H•).

i

Because H is the pullback of a function on Q, its derivative is of the form

dH = Hν θ + Hi•i ,

and using this and the structure equations (4.14), one can verify that Π is closed.

Because Π is semibasic over M and closed, it is the pullback of a closed form on

M , which is then a de¬nite, neo-classical Poincar´-Cartan form. The associated

e

Euler-Lagrange di¬erential system then pulls back to F as

EH = {θ, dθ, •0 § •(i) ’ H•}.

i

While (4.15) shows that generic Legendre n-planes in M are de¬ned by equations

•0 ’ hij •j = 0,

θ = 0, i

with hij = hji, integral n-planes in M for EH are de¬ned by the same equations,

plus

hii = H.

The functions hij describing the tangent locus of a transverse Legendre subman-

ifold of M are of course the coe¬cients of the second fundamental form of the

corresponding submanifold of Q. Therefore, the transverse integral manifolds

of EH correspond locally to hypersurfaces in Q whose mean curvature hii equals

the background function H. This will appear quite explicitly in what follows.

To investigate these integral manifolds, we employ the following apparatus.

First consider the product

F — Rn(n+1)/2,

where Rn(n+1)/2 has coordinates hij = hji, and inside this product de¬ne the

locus

def

F (1) = (f, h) ∈ F — Rn(n+1)/2 : hii = H .

To perform calculations, we want to extend our parallelization of F to F (1) .

With a view toward reconstructing some of the bundle B ’ M associated to

the Poincar´-Cartan form Π, we do this in a way that is as well-adapted to Π

e

as possible.

On F (1), we continue to work with θ, and de¬ne

def

ω i = •i ,

def

πi = •0 ’ hij •j .

i

144 CHAPTER 4. ADDITIONAL TOPICS

With these de¬nitions, we have

dθ = ’πi § ωi ,

Π = ’θ § πi § ω(i).

Motivated by Riemannian geometry, we set

def

Dhij = dhij ’ hkj ±k ’ hik ±k ,

i j

so that in particular Dhij = Dhji and Dhii = dH. We also de¬ne the traceless

part

def

Dh0 = Dhij ’ n δij dH.

1

ij

Direct computations show that we will have exactly the structure equations (4.2,

4.3, 4.4) if we de¬ne

±i = •i ,

j j

1

= ’ 2n Hθ,

ρ

= ’ 1 Hπi + hij πj + (hik hkj ’ R0 ’ n δij Hν )ωj ,

1

δi ij0

2

σij § ωj = (Dh0 + n δij Hk ωk + 1 R0 ωk ) § ωj ,

1

ij 2 ijk

with σij = σji, σii = 0. The last item requires some comment. Some linear

algebra involving a Koszul complex shows that for any tensor Vijk with Vijk =

’Vikj (this will be applied to 2n (δij Hk ’ δik Hj ) + 1 R0 ), there is another

1

2 ijk

tensor Wijk , not unique, satisfying Wijk = Wjik, Wiik = 0, and 1 (Wijk ’

2

Wikj ) = Vijk. This justi¬es the existence of σij satisfying our requirements.

The structure equations (4.2, 4.3, 4.4) resulting from these assignments have

torsion coe¬cients

T ijk = 0, U ij = ’δ ij , i 1i

Sj = ’hij + n δj H.

The general calculations of §4.1.1 for the second variation can now be applied;

note that we have the freedom to adapt coframes to a single integral submanifold

in M of EH . Repeating those calculations verbatim leads to

d2

g(dgi + nρgi ’ gj ±j + gδi ) § ω(i) ,

=’

Λ i

dt2 Nt N0

t=0

where F : N — [0, 1] ’ M is a Legendre variation in M , F0 is an integral

manifold of EH , and the forms are all pullbacks of forms on F — (F (1)) by a

section of F — (F (1) ) ’ N — [0, 1], adapted along N0 in the sense that

θN0 = g dt, (πi )N0 = gidt.

These imply that restricted to N0 , we have θ|N0 = πi |N0 = 0, and the preceding

formula becomes

g(dgi ’ gj ±j + g(hik hkj ’ n δij Hν ’ R0 )ωj ) § ω(i) .

δ 2 (FΛ )N0 (g) = ’ 1

ij0

i

N0

4.1. THE SECOND VARIATION 145

Recognizing that g can be thought of as a section of the normal bundle of the

F0 q

hypersurface N ’ M ’ Q, and that in this case gi are the coe¬cients of its

covariant derivative, this can be rewritten as

δ 2 (FΛ)N0 (g) = ’ (g∆g + g2 (||h||2 ’ Hν ’ R0 ))ω, (4.16)

ii0

N0

where ∆ is the Riemannian Laplacian, and ||h||2 = T r(h— h). Here, the extrinsic

curvature function K appears as the quantity ||h||2 ’ Hν ’ R0 . Notice that

ii0

we have actually calculated this second variation without ever determining the

functional Λ. In case the ambient manifold Q is ¬‚at Euclidean space, if the

background function H is a constant and the variation g is compactly supported

in the interior of N , this simpli¬es to

d2

(g∆g + g2 ||h||2)ω

=’

Λ

dt2 Nt N0

t=0

(|| g||2 ’ g2 ||h||2)ω.

=

N0

Even for the minimal surface equation H = 0, we cannot conclude from this

formula alone that a solution locally minimizes area.

4.1.5 Conditions for a Local Minimum

We now discuss some conditions under which an integral manifold N ’ M of

an Euler-Lagrange system EΛ ‚ „¦— (M ) is a local minimum for the functional

FΛ , in the sense that FΛ (N ) < FΛ(N ) for all Legendre submanifolds N near

N . However, there are two natural meanings for “near” in this context, and

this will yield two notions of local minimum. Namely, we will say that FΛ has

a strong local minimum at N if the preceding inequality holds whenever N

is C 0-close to N , while FΛ has a weak local minimum at N if the preceding

inequality holds only among the narrower class of N which are C 1-close to N .2

Our goal is to illustrate how the Poincar´-Cartan form may be used to un-

e

derstand in a simple geometric manner some classical conditions on extrema.

Speci¬cally, we will introduce the notion of a calibration for an integral manifold

of the Euler-Lagrange system; its existence (under mild topological hypotheses)

implies that the integral manifold is a strong local minimum. Under certain

classical conditions for a local minimum, we will use the Poincar´-Cartan form

e

to construct an analogous weak calibration. Finally, our geometric description of

the second variation formula highlights the Jacobi operator Jg = ’∆c g + Kgω,

and some linear analysis shows that the positivity of the ¬rst eigenvalue of J

implies the classical conditions.

Let Π be a neo-classical Poincar´-Cartan form Π on a contact manifold

e

(M, I). There is a local foliation M ’ Q, and we can choose coordinates to

2A thorough, coordinate-based discussion of the relevant analysis can be found in [GH96].

146 CHAPTER 4. ADDITIONAL TOPICS

have (xi, z) ∈ Q ‚ Rn — R, (xi, z, pi) ∈ M ‚ J 1 (Rn, R), θ = dz ’ pi dxi ∈ “(I),

and a Lagrangian potential

Λ = L(x, z, p)dx + θ § Lpi dx(i) ∈ „¦n (M )

whose Poincar´-Cartan form is

e

Π = dΛ = θ § (’dLpi § dx(i) + Lz dx).

We will con¬ne our discussion to a domain where this classical description holds.

We may regard an integral manifold of the Euler-Lagrange system EΛ as a

submanifold N0 ’ Q given by the graph {(x, z0(x)) : x ∈ U } of a solution to

the Euler-Lagrange equation over some open U ‚ Rn. It has a natural 1-jet

(1)

extension N0 ’ M , equal to {(x, z0(x), z0(x)) : x ∈ U }, which is an integral

manifold of EΛ in the sense discussed previously. We de¬ne a strong neighborhood

of N0 to be the collection of hypersurfaces in Q lying in some open neighborhood

of N0 in Q, and a weak neighborhood of N0 to be the collection of hypersurfaces

(1)

N ’ Q whose 1-jet prolongations N (1) lie in some open neighborhood of N0

in M . Whether or not a given stationary submanifold N0 is minimal depends

on which of these two classes of competing submanifolds one studies.

Starting with strong neighborhoods, we ¬x a neighborhood W ‚ Q of a

stationary submanifold N0 ‚ Q for Λ, and introduce the following useful notion.

˜

De¬nition 4.1 A calibration for (Λ, N0) is an n-form Λ ∈ „¦n(W ) satisfying

˜