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 ā„ 3, J k = ā„¦k (Cn ), and that the integral 2-planes in T Cn
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
Ļ• ā ā„¦1,0 (Cn ), we ļ¬nd that
dĻ• + dĻ• ā J ,
32 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS
so that Ļ• + Ļ• is a conservation law for J . These give rise to inļ¬nitely many
which must be satisļ¬ed by Ī³C , 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 Ļ• = df for some
holomorphic function f ā O(Cn ). Another fact, not to be proved here, is that
these moment conditions are suļ¬cient for Ī³C to bound a (possibly branched)
Returning to our discussion of minimal surfaces, suppose that x : U ā’ E3 is
a minimal immersion of a simply connected surface. Then ā—dx deļ¬nes a closed,
vector-valued 1-form on U , so there exists a vector-valued function y : U ā’ E 3
dy = ā—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 deļ¬ne
z = (x + iy) : U ā’ 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 Ī³ (1) , the Euclidean moment condition
ā—dx = 0 implies that there exists another real curve y so that dy = ā—dx (along
Ī³). Then we can use z = (x + iy) : Ī³ ā’ 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 ļ¬ll the
real curve Ī³ 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 PoincarĀ“-Cartan form
Ī H = ā’Īø ā§ Ļi ā§ Ļ(i) ā’ HĻ
is invariant under the full Euclidean group, but that no particular Lagrangian
Ī is so invariant; we will continue to work with the SO(n + 1, R)-invariant
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 33
ĪH = Ļ + x ā„¦, dĪH = Ī H .
Fortunately, the equivalence class [Ī] ā H n(ā„¦ā— (M )/I) is invariant under the
Euclidean group, because as the reader can verify, the connecting map
Ī“ : H n(ā„¦ā— /I) ā’ H n+1(I)
taking [Ī] to Ī is an isomorphism for this contact manifold. This means that,
as in the case H = 0, we will ļ¬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 ļ¬elds v ā Rn+1 which fail to preserve our ĪH . Instead, we have
Lv ĪH n+1 Lv (x ā„¦) (because Lv Ļ =
= ((Lv x) ā„¦ + x (Lv ā„¦))
= n+1 (v ā„¦ + 0).
In the last step, we have used Lv x = [v, x] = v (by a simple calculation), and
Lv ā„¦ = 0 (because the ambient volume ā„¦ is translation invariant). To apply the
Noether prescription, we need an anti-derivative of this last term, which we ļ¬nd
Lx (v ā„¦) ā’ x d(v ā„¦)
d(x (v ā„¦)) =
((Lx v) ā„¦ + v (Lxā„¦)) ā’ x
ā’v ā„¦ + (n + 1)v ā„¦,
where we have again used Lxv = [x, v] = ā’v, and Lxā„¦ = (n + 1)ā„¦. Combining
these two calculations, we have
Lv ĪH = d(x (v ā„¦)).
The prescription (1.13, 1.14, 1.15) now gives
Ļ•v = ĪH + x (v ā„¦)
= Ļ+ nx (v ā„¦).
As in the case of minimal hypersurfaces, we consider the restriction of Ļ•v to an
integral manifold N . From the previous case, we know that v Ļ restricts to
ā— v, dx , where ā— = ā—N is the star operator of the metric on N and Ā·, Ā· denotes
the ambient inner-product. To express the restriction of the other term of Ļ•v ,
decompose x = xt + xĪ½ Ī½ into tangential and normal parts along N (so xt is a
vector and xĪ½ is a scalar), and a calculation gives
ā’ H (xĪ½ v Ļ ā’ (v
nx (v ā„¦)|N = Īø)(xt Ļ))
ā’ H (xĪ½ ā— v, dx ā’ (v Īø) ā— Ā·, xt );
34 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS
the latter ā— 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 ā’ 1)-forms
Ļ•v , which depend linearly on v ā Rn+1 , as an (Rn+1 )ā— -valued (n ā’ 1)-form on
N . It is
Ļ•trans = ā’(1 + H xĪ½ ) ā—dx + H Ī½ ā— Ā·, xt .
In the second term, the normal Ī½ provides the āvector-valuedā part (it replaced
Īø, to which it is dual), and ā— Ā·, xt provides the ā(n ā’ 1)-formā part.
Calculating the conservation laws for rotations is a similar process, simpliļ¬ed
somewhat by the fact that Lv ĪH = 0; of course, the lifted rotation vector ļ¬elds
v are not so easy to work with as the translations. The resulting Ī2 Rn+1-valued
(n ā’ 1)-form is
Ļ•rot = ā’(1 + ā—(x ā§ dx) + ā§ Ī½) ā— Ā·, xt .
n+1 xĪ½ ) 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 EĪ might have an inļ¬nitesimal symmetry which scales the corresponding
PoincarĀ“-Cartan form Ī . This is the case for the minimal surface system, which
is preserved by the dilation vector ļ¬eld on En+1
xa ā‚xa .
This induces a vector ļ¬eld x on the contact manifold of tangent hyperplanes
to En+1 where the functional Ī and PoincarĀ“-Cartan form Ī are deļ¬ned, and
there are various ways to calculate that
Lx Ī = nĪ.
If one tries to apply the Noether prescription to x by writing
Ļ•dil = x Ī,
the resulting form satisļ¬es
LxĪ ā’ x dĪ
nĪ ā’ x Ī .
Restricted to a minimal surface N , we will then have
dĻ•dil |N = nĪ.
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 ļ¬nd
Ļ•=n Ī. (1.22)
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 ļ¬eld (e0 , . . . , en) along
U āŖ ā‚U satisfying the conditions (1.20) as before. We write the coeļ¬cients
xa ā‚xa = v a ea ,
and then restricted to ā‚U , we have
Ī = (ā’1)nā’1vn Ļ1 ā§ Ā· Ā· Ā· ā§ Ļnā’1.
Ļ•|ā‚U = x
Up to sign, the form Ļ 1 ā§ Ā· Ā· Ā·ā§ Ļnā’1 along ā‚U is exactly the (n ā’ 1)-dimensional
area form for ā‚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, deļ¬ned as the intersection of
N ā‚ En+1 with an origin-centered ball of radius r. In particular, along ā‚Ur we
will have ||x|| = r, so that v n ā¤ r and
r Ā· Area(ā‚Ur ) ā„ n Ā· Vol(Ur ). (1.23)
Area(ā‚Ur ) = Vol(Ur ),
and (1.23) is now a diļ¬erential inequality for Vol(Ur ) which can be solved to
Vol(Ur ) ā„ Crn
for some constant C. This is a remarkable result about minimal hypersurfaces,
and amply illustrates the power of āalmost-conservation lawsā like Ļ•dil .
36 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS
The Geometry of
In this chapter, we will study some of the geometry associated to PoincarĀ“- e
Cartan forms using E. Cartanā™s method of equivalence. The idea is to identify
such a PoincarĀ“-Cartan form with a G-structureā”that is, a subbundle of the
principal coframe bundle of a manifoldā”and then attempt to ļ¬nd some canon-
ically determined basis of 1-forms on the total space of that G-structure. The
diļ¬erential structure equations of these 1-forms will then exhibit associated ge-
ometric objects and invariants.
The pointwise linear algebra of a PoincarĀ“-Cartan form in the case of n = 2
āindependent variablesā (that is, on a contact manifold of dimension 5) is quite
diļ¬erent from that of higher dimensional cases. Therefore, in the ļ¬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 PoincarĀ“- 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 ļ¬rst identify in case n ā„ 3 a narrower class
of PoincarĀ“-Cartan forms, called neo-classical, which are of the same algebraic
type as those arising from classical variational problems. We will describe some
of the geometry associated with neo-classical PoincarĀ“-Cartan forms, consisting
of a ļ¬eld of hypersurfaces in a vector bundle, well-deļ¬ned up to ļ¬berwise aļ¬ne
motions of the vector bundle. A digression on the local geometry of individual
hypersurfaces in aļ¬ne space follows this. We then turn to the very rich equiva-
lence problem for neo-classical PoincarĀ“-Cartan forms; the diļ¬erential invariants
that this uncovers include those of the various associated aļ¬ne hypersurfaces.
In the last section of this chapter, we use these diļ¬erential invariants to charac-
terize systems locally contact-equivalent to those for prescribed mean curvature
38 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS