˜

• Λ|N0 = Λ|N (1) ;

0

˜

• Λ|E ¤ ΛE for each n-plane E n ‚ Tq W .

In the right-hand side of the last inequality, we are regarding the n-plane E

as specifying a point of M ‚ Gn(T Q) over q ∈ Q, and evaluating Λ(q,E) on

any tangent n-plane E ‚ T(q,E) M projecting one-to-one into Tq Q; the value is

independent of the choice of E , because Λ is semibasic over Q. In particular,

˜

the third condition says that the integral of Λ on any N ‚ W ‚ Q will not

(1)

exceed the integral of Λ on N ‚ M .

In both the strong and weak settings, we will only have N0 compete against

submanifolds having the same boundary. For this reason, we assume that W ⊃

N0 is chosen so that ‚N0 = N0 © ‚W , and that (N0 , ‚N0 ) generates the relative

homology Hn (W, ‚W ; Z).

˜

Proposition 4.1 If there exists a calibration Λ for (Λ, N0), then FΛ (N0 ) ¤

FΛ (N ) for every hypersurface N ’ W ‚ Q satisfying ‚N = ‚N0 .

We then say that N0 is a strong (but not strict!) local minimum for FΛ .

4.1. THE SECOND VARIATION 147

Proof. We simply calculate

FΛ (N0 ) = Λ

(1)

N0

˜

= Λ

N0

˜

= Λ

N

¤ Λ

N (1)

FΛ (N ).

=

The third equality uses Stokes™ theorem, which applies because our topological

hypothesis on W implies that the cycle N ’ N0 in W is a boundary.

The question of when one can ¬nd a calibration naturally arises. For this,

we use the following classical concept.

De¬nition 4.2 A ¬eld for (Λ, N0 ) is a neighborhood W ‚ Q of N0 with a

smooth foliation by a 1-parameter family F : N — (’µ, µ) ’ W of integral

manifolds of EΛ .

This family does not have a ¬xed boundary. We retain the topological hypothe-

ses on W used in Proposition 4.1, and have the following.

Proposition 4.2 If there exists a ¬eld for (Λ, N0), then there exists a closed

˜ ˜

form Λ ∈ „¦n (W ) such that Λ|N0 = Λ|N (1) .

0

˜ ˜

Λ is then a calibration if it additionally satis¬es the third condition, Λ|E ¤ ΛE .

˜

In the proof, we will explicitly construct Λ using the Poincar´-Cartan form.

e

Proof. The ¬eld F : N — (’µ, µ) ’ Q may be thought of as a family of graphs

Nt = {(x, z(x, t))},

where each z(·, t) is a solution of the Euler-Lagrange equations, and the domain

of z(·, t) may depend on t ∈ (’µ, µ). Because each point of W lies on exactly

one of these graphs, we can de¬ne a 1-jet lift F : W ’ M , given by

(x, z) = (x, z(x, t)) ’ (x, z(x, t), xz(x, t)).

˜ ˜

Let Λ = (F )— Λ ∈ „¦n (W ). Then it is clear that Λ|N0 = Λ|N (1) , and to show that

0

˜ ˜

Λ is closed, we need to see that (F )— Π = dΛ = 0. This holds because Π = θ § Ψ

(1)

is quadratic in an ideal of forms vanishing on each leaf Ft : N ’ M ; more

concretely, each of (F )— θ and (F )— Ψ must be a multiple of dt, so their product

vanishes.

148 CHAPTER 4. ADDITIONAL TOPICS

˜

General conditions for Λ = (F )— Λ to be a calibration, and for N0 to therefore

be a strong local minimum, are not clear. However, we can still use the preceding

to detect weak local minima.

Proposition 4.3 Under the hypotheses of Propositions 4.1 and 4.2, if

z0(x))ξi ξj ≥ c||ξ||2,

Lpi pj (x, z0 (x), (4.17)

˜

for some constant c > 0 and all (ξi ), then Λ|E ¤ ΛE for all E ‚ Tq Q su¬ciently

near Tq N0 , with equality if and only if E = Tq N0 . Furthermore, FΛ(N0 ) <

FΛ (N ) for all N = N0 in a weak neighborhood of N0 .

˜

The ¬rst statement allows us to think of Λ as a weak calibration for (Λ, N0).

The proof of the second statement from the ¬rst will use Stokes™ theorem in

exactly the manner of Proposition 4.1.

Proof. The positivity of the z-Hessian of L suggests that we de¬ne the

Weierstrass excess function

def

E(x, z, p, q) = L(x, z, p) ’ L(x, z, q) ’ (pi ’ qi )Lqi (x, z, q),

which is the second-order remainder in a Taylor series expansion for L. This

˜

function will appear in a more detailed expression for π — Λ = (F —¦ π)— Λ ∈

„¦n (M ), computed modulo {I}. We write

F (xi , z) = (xi, z, qi(x, z)) ∈ M,

where (xi , z, pi) are the usual coordinates on M , and the functions qi(x, z) are

the partial derivatives of the ¬eld elements z(x, t). We have

˜

π— Λ = π— —¦ F — (L dx + θ § Lpi dx(i))

L(xi , z, qi(x, z))dx + (dz ’ qi(x, z)dxi) § Lpi (xi, z, qi(x, z))dx(i)

=

L(xi , z, qi(x, z)) + (pi ’ qi (x, z))Lpi (xi , z, qi(x, z)) dx (mod {I})

≡

Λ ’ E(xi, z, pi, qi(x, z))dx (mod {I}).

≡

The hypothesis (4.17) on the Hessian Fpi pj implies that for each (xi, z), and pi

su¬ciently close to qi (x, z), the second-order remainder satis¬es

E(xi, z, pi, qi(x, z)) ≥ 0,

˜

with equality if and only if pi = qi(x, z). The congruence of π — Λ and Λ ’ E dx

modulo {I} then implies our ¬rst statement.

4.1. THE SECOND VARIATION 149

For the second statement, we use the Stokes™ theorem argument:

˜

Λ= Λ

(1)

N0 N0

˜

= Λ

N

(Λ ’ E(xi , z, pi, qi(x, z))dx)

=

N (1)

¤ Λ,

N (1)

with equality in the last step if and only if N0 = N .

This proof shows additionally that if the Weiestrass excess function satis¬es

E(xi , z, pi, qi) > 0 for all p = q, then N0 is a strong (and strict) local minimum

for FΛ.

So far, we have shown that if we can cover some neighborhood of a sta-

˜

tionary submanifold N0 with a ¬eld, then we can construct an n-form Λ, whose

calibration properties imply extremal properties of N0. It is therefore natural to

ask when there exists such a ¬eld, and the answer to this involves some analysis

of the Jacobi operator. We will describe the operator, and hint at the analysis.

The Jacobi operator acts on sections of a density line bundle on a given

integral manifold N0 of the Euler-Lagrange system, with its induced conformal

structure. Speci¬cally,

n’2 n+2

J : D 2n ’ D 2n

is the di¬erential operator given by

Jg = ’∆c g ’ gK,

where ∆c is the conformal Laplacian, and K is the curvature invariant intro-

duced in §4.1.2. The second variation formula (4.8) then reads

δ 2 (FΛ)N0 (g) = g Jg ω.

N0

The main geometric fact is:

The Jacobi operator gives the (linear) variational equations for in-

tegral manifolds of the Euler-Lagrange system EΛ .

This means the following. Let F : N —[0, 1] ’ M be a Legendre variation of the

Λ-stationary submanifold F0”not necessarily having ¬xed boundary”and let

g = ‚F |t=0 θ, as usual. Then our previous calculations imply that Jg = 0

‚t

if and only if

L ‚ (F — Ψ)|t=0 = 0.

‚t

We might express condition by saying that Ft is an integral manifold for EΛ =

I + {Ψ} modulo O(t2 ).

150 CHAPTER 4. ADDITIONAL TOPICS

We now indicate how a condition on the Jacobi operator of N0 can imply

the existence of a ¬eld near N0 . Consider the eigenvalue problem

∞

Jg = ’∆c g ’ gK = »g, g ∈ C0 (N ),

for smooth, ¬xed boundary variations. It is well-known J has a discrete spec-

trum bounded from below, »1 < »2 < · · · , with »k ’ ∞ and with ¬nite-

dimensional eigenspaces. We consider the consequences of the assumption

»1 > 0.

g Jg ω : ||g||L2 = 1}, the assumption »1 > 0 is equivalent

Because »1 = inf{ N

to

δ 2 (FΛ )N (g) > 0, for g = 0.

The main analytic result is the following.

Proposition 4.4 If »1 > 0, then given g0 ∈ C ∞ (‚N ), there is a unique solu-

tion g ∈ C ∞ (N ) to the boundary value problem

Jg = 0, g|‚N = g0.

Furthermore, if g0 > 0 on ‚N , then this solution satis¬es g > 0 on N .

The existence and uniqueness statements follow from standard elliptic theory.

The point is that we can compare the second variation δ 2(FΛ )N (g) to the

Sobolev norm ||g||2 = N (|| g||2+|g|2 )ω, and if »1 > 0, then there are constants

1

c1 , c2 > 0 such that

c1||g||2 ¤ g Jg ω ¤ c2 ||g||2.

1 1

N

The Schauder theory gives existence and uniqueness in this situation.

Less standard is the positivity of the solution g under the assumption that

g|‚N > 0, and this is crucial for the existence of a ¬eld. Namely, a further

implicit function argument using elliptic theory guarantees that the variation g

is tangent to an arc of integral manifolds of E, and the fact that g = 0 implies

that near the initial N , this arc de¬nes a ¬eld. For the proof of the positivity of

g, and details of all of the analysis, see Giaquinta & Hildebrandt (cit. p. 145n).

4.2 Euler-Lagrange PDE Systems

Up to this point, we have studied geometric aspects of ¬rst-order Lagrangian

functionals

L xi , z, ‚xi dx, „¦ ‚ Rn ,

‚z

FL (z) = (4.18)

„¦

4.2. EULER-LAGRANGE PDE SYSTEMS 151

where x = (x1, . . . , xn) and z = z(x) is a scalar function. In this section, we

consider the more general situation of functionals

±

L xi , z ±(x), ‚z i (x) dx, „¦ ‚ Rn ,

FL(z) = (4.19)

‚x

„¦

where now z(x) = (z 1 (x), . . . , z s(x)) is an Rs-valued function of x = (xi ),

and L = L(xi , z ±, p±) is a smooth function on Rn+s+ns. The Euler-Lagrange

i

equations describing maps z : „¦ ’ Rs which are stationary for FL under all

¬xed-boundary variations form a PDE system