R

In other words, J is algebraically generated by real 2-forms which, when re-

garded as complex 2-forms, have no part of type (1, 1). It is elementary to see

that in degree k ≥ 3, J k = „¦k (Cn ), and that the integral 2-planes in T Cn

R

are exactly the complex 1-dimensional subspaces. This implies our claim that

integral manifolds of J are holomorphic curves.

Now, J has many conservation laws. Namely, for any holomorphic 1-form

• ∈ „¦1,0 (Cn ), we ¬nd that

hol

d• + d• ∈ J ,

¯

´

32 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

so that • + • is a conservation law for J . These give rise to in¬nitely many

¯

moment conditions

•=0

γC

which must be satis¬ed by γC , if it is to be the boundary of a holomorphic disc.

It is a fact which we shall not prove here that every conservation law for

J is of this form; trivial conservation laws clearly arise when • = df for some

holomorphic function f ∈ O(Cn ). Another fact, not to be proved here, is that

these moment conditions are su¬cient for γC to bound a (possibly branched)

holomorphic disc.

Returning to our discussion of minimal surfaces, suppose that x : U ’ E3 is

a minimal immersion of a simply connected surface. Then —dx de¬nes a closed,

vector-valued 1-form on U , so there exists a vector-valued function y : U ’ E 3

satisfying

dy = —dx. (1.21)

Note that our ability to integrate the conservation law to obtain a function relies

essentially on the fact that we are in dimension n = 2.

We can de¬ne

z = (x + iy) : U ’ C3 ,

and (1.21) is essentially the Cauchy-Riemann equations, implying that z is a

holomorphic curve, with the conformal structure induced from the immersion

z. Furthermore, the complex derivative z is at each point of U a null vector

for the complex bilinear inner-product (dz i)2 . This gives the classical Weier-

strass representation of a minimal surface in E3 as locally the real part of a

holomorphic null curve in C3 .

We can now incorporate the result of our digression on conservation laws for

holomorphic discs. Namely, given a strip γ (1) , the Euclidean moment condition

—dx = 0 implies that there exists another real curve y so that dy = —dx (along

γ

γ). Then we can use z = (x + iy) : γ ’ C3 as initial data for the holomorphic

disc problem, and all of the holomorphic moment conditions for that problem

come into play. These are the additional hidden constraints needed to ¬ll the

real curve γ with a (possibly branched) minimal surface.

Conservation Laws for Constant Mean Curvature

It is also a worthwhile exercise to determine the conservation laws corresponding

to Euclidean motions for the constant mean curvature system when the constant

H is non-zero. Recall that for that system the Poincar´-Cartan form

e

ΠH = ’θ § πi § ω(i) ’ Hω

is invariant under the full Euclidean group, but that no particular Lagrangian

Λ is so invariant; we will continue to work with the SO(n + 1, R)-invariant

1.4. HYPERSURFACES IN EUCLIDEAN SPACE 33

Lagrangian

H

ΛH = ω + x „¦, dΛH = ΠH .

n+1

Fortunately, the equivalence class [Λ] ∈ H n(„¦— (M )/I) is invariant under the

Euclidean group, because as the reader can verify, the connecting map

δ : H n(„¦— /I) ’ H n+1(I)

taking [Λ] to Π is an isomorphism for this contact manifold. This means that,

as in the case H = 0, we will ¬nd conservation laws corresponding to the full

Euclidean Lie algebra.

Computing the conservation laws corresponding to translations requires the

more complicated form of the Noether prescription, because it is the translation

vector ¬elds v ∈ Rn+1 which fail to preserve our ΛH . Instead, we have

H

Lv ΛH n+1 Lv (x „¦) (because Lv ω =

= 0),

H

= ((Lv x) „¦ + x (Lv „¦))

n+1

H

= n+1 (v „¦ + 0).

In the last step, we have used Lv x = [v, x] = v (by a simple calculation), and

Lv „¦ = 0 (because the ambient volume „¦ is translation invariant). To apply the

Noether prescription, we need an anti-derivative of this last term, which we ¬nd

by experimenting:

Lx (v „¦) ’ x d(v „¦)

d(x (v „¦)) =

((Lx v) „¦ + v (Lx„¦)) ’ x

= 0

’v „¦ + (n + 1)v „¦,

=

where we have again used Lxv = [x, v] = ’v, and Lx„¦ = (n + 1)„¦. Combining

these two calculations, we have

H

Lv ΛH = d(x (v „¦)).

n(n+1)

The prescription (1.13, 1.14, 1.15) now gives

H

’v

•v = ΛH + x (v „¦)

n(n+1)

H

’v

= ω+ nx (v „¦).

As in the case of minimal hypersurfaces, we consider the restriction of •v to an

integral manifold N . From the previous case, we know that v ω restricts to

— v, dx , where — = —N is the star operator of the metric on N and ·, · denotes

the ambient inner-product. To express the restriction of the other term of •v ,

decompose x = xt + xν ν into tangential and normal parts along N (so xt is a

vector and xν is a scalar), and a calculation gives

H

’ H (xν v ω ’ (v

nx (v „¦)|N = θ)(xt ω))

n

’ H (xν — v, dx ’ (v θ) — ·, xt );

= n

´

34 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

the latter — is being applied to the 1-form on N that is dual via the metric to the

tangent vector xt . Again as in the H = 0 case, we can write these (n ’ 1)-forms

•v , which depend linearly on v ∈ Rn+1 , as an (Rn+1 )— -valued (n ’ 1)-form on

N . It is

•trans = ’(1 + H xν ) —dx + H ν — ·, xt .

n n

In the second term, the normal ν provides the “vector-valued” part (it replaced

θ, to which it is dual), and — ·, xt provides the “(n ’ 1)-form” part.

Calculating the conservation laws for rotations is a similar process, simpli¬ed

somewhat by the fact that Lv ΛH = 0; of course, the lifted rotation vector ¬elds

v are not so easy to work with as the translations. The resulting Λ2 Rn+1-valued

(n ’ 1)-form is

H H

•rot = ’(1 + —(x § dx) + § ν) — ·, xt .

n+1 xν ) n+1 (x

These can be used to produce moment conditions, just as in the H = 0 case.

We conclude with one more observation suggesting extensions of the notion

of a conservation law. Recall that we showed in (1.7) that a Monge-Ampere

system EΛ might have an in¬nitesimal symmetry which scales the corresponding

Poincar´-Cartan form Π. This is the case for the minimal surface system, which

e

is preserved by the dilation vector ¬eld on En+1

xa ‚xa .

‚

x=

This induces a vector ¬eld x on the contact manifold of tangent hyperplanes

to En+1 where the functional Λ and Poincar´-Cartan form Π are de¬ned, and

e

there are various ways to calculate that

Lx Λ = nΛ.

If one tries to apply the Noether prescription to x by writing

•dil = x Λ,

the resulting form satis¬es

LxΛ ’ x dΛ

d•dil =

nΛ ’ x Π.

=

Restricted to a minimal surface N , we will then have

d•dil |N = nΛ.

Because the right-hand side is not zero, we do not have a conservation law, but

it is still reasonable to look for consequences of integrating on neighborhoods U

in N , where we ¬nd

•=n Λ. (1.22)

‚U U

1.4. HYPERSURFACES IN EUCLIDEAN SPACE 35

The right-hand side equals n times the area of U , and the left-hand side can be

investigated by choosing an oriented orthonormal frame ¬eld (e0 , . . . , en) along

U ∪ ‚U satisfying the conditions (1.20) as before. We write the coe¬cients

xa ‚xa = v a ea ,

‚

x=

and then restricted to ‚U , we have

Λ = (’1)n’1vn ω1 § · · · § ωn’1.

•|‚U = x

Up to sign, the form ω 1 § · · ·§ ωn’1 along ‚U is exactly the (n ’ 1)-dimensional

area form for ‚U .

These interpretations of the two sides of (1.22) can be exploited by taking

for U the family of neighborhoods Ur for r > 0, de¬ned as the intersection of

N ‚ En+1 with an origin-centered ball of radius r. In particular, along ‚Ur we

will have ||x|| = r, so that v n ¤ r and

r · Area(‚Ur ) ≥ n · Vol(Ur ). (1.23)

Observe that

d

Area(‚Ur ) = Vol(Ur ),

dr

and (1.23) is now a di¬erential inequality for Vol(Ur ) which can be solved to

give

Vol(Ur ) ≥ Crn

for some constant C. This is a remarkable result about minimal hypersurfaces,

and amply illustrates the power of “almost-conservation laws” like •dil .

´

36 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Chapter 2

The Geometry of

Poincar´-Cartan Forms

e

In this chapter, we will study some of the geometry associated to Poincar´- e

´

Cartan forms using E. Cartan™s method of equivalence. The idea is to identify

such a Poincar´-Cartan form with a G-structure”that is, a subbundle of the

e

principal coframe bundle of a manifold”and then attempt to ¬nd some canon-

ically determined basis of 1-forms on the total space of that G-structure. The

di¬erential structure equations of these 1-forms will then exhibit associated ge-

ometric objects and invariants.

The pointwise linear algebra of a Poincar´-Cartan form in the case of n = 2

e

“independent variables” (that is, on a contact manifold of dimension 5) is quite

di¬erent from that of higher dimensional cases. Therefore, in the ¬rst section we

study only the former, which should serve as a good illustration of the method

of equivalence for those not familiar with it. Actually, in case n = 2 we will

study the coarser equivalence of Monge-Ampere systems rather than Poincar´- e

Cartan forms, and we will do this without restricting to those systems which are

locally Euler-Lagrange. An extensive study of the geometry of Monge-Ampere

systems in various low dimensions was carried out in [LRC93], with a viewpoint

somewhat similar to ours.

In the succeeding sections, we will ¬rst identify in case n ≥ 3 a narrower class

of Poincar´-Cartan forms, called neo-classical, which are of the same algebraic

e

type as those arising from classical variational problems. We will describe some

of the geometry associated with neo-classical Poincar´-Cartan forms, consisting

e

of a ¬eld of hypersurfaces in a vector bundle, well-de¬ned up to ¬berwise a¬ne

motions of the vector bundle. A digression on the local geometry of individual

hypersurfaces in a¬ne space follows this. We then turn to the very rich equiva-

lence problem for neo-classical Poincar´-Cartan forms; the di¬erential invariants

e

that this uncovers include those of the various associated a¬ne hypersurfaces.

In the last section of this chapter, we use these di¬erential invariants to charac-

terize systems locally contact-equivalent to those for prescribed mean curvature

37

´

38 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS