Restricted to a transverse Legendre submanifold over a surface N ‚ E3 , these

give K dA, H dA, and the area form dA of N , respectively. Linear Weingarten

surfaces are integral manifolds of a Monge-Ampere system

def

{θ, dθ, Ψ(a, b, c) = aΨ2 + bΨ1 + cΨ0 }.

Our previous Lie derivative computations may be used to compute

« « «

0 ’1 0

Ψ0 Ψ0

Lv Ψ1 = 0 0 ’2 Ψ1 .

Ψ2 00 0 Ψ2

Exponentiate this to see

•— Ψ(a, b, c) = Ψ(a ’ 2bt + ct2, b ’ ct, c). (1.18)

t

This describes how the 1-parameter group •t acts on the collection of linear

Weingarten equations. Similar calculations show that the 1-parameter group

ψs introduced earlier consists of contact transformations, and acts on linear

Weingarten equations as

—

ψs Ψ(a, b, c) = Ψ(a, exp(s)b, exp(2s)c). (1.19)

It is reasonable to regard the coe¬cients (a, b, c) which specify a particular

linear Weingarten equation as a point [a : b : c] in the real projective plane RP2 ,

and it is an easy exercise to determine the orbits in RP2 of the group action

generated by (1.18) and (1.19). There are ¬ve orbits, represented by the points

[1 : 0 : 0], [0 : 1 : 0], [1 : 0 : 1], [1 : 0 : ’1], [0 : 0 : 1]. The special case

•—1 Ψ(0, 1, A) = Ψ(’ A , 0, A)

1

A

gives the classically known fact that to every surface of non-zero constant mean

curvature ’A, there is a (possibly singular) parallel surface of constant positive

Gauss curvature A2 . Note ¬nally that the Monge-Ampere system corresponding

to [0 : 0 : 1] has for integral manifolds those non-transverse Legendre submani-

folds of M which project to curves in E3, instead of surfaces.

1.4.3 Conservation Laws for Minimal Hypersurfaces

In Chapter 3, we will be concerned with conservation laws for various Euler-

Lagrange equations arising in conformal geometry. We will emphasize two ques-

tions: how are conservation laws found, and how can they be used? In this

section, we will explore these two questions in the case of the minimal hyper-

surface equation H = 0, regarding conservation laws arising from Euclidean

symmetries.

We compute these conservation laws ¬rst for the translations, and then for

the rotations. The results of these computations will be the two vector-valued

conservation laws

d(—(x § dx)) = 0.

d(—dx) = 0,

´

28 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

The notation will be explained in the course of the calculation. These may be

thought of as analogs of the conservation of linear and angular momentum that

are ubiquitous in physics.

To carry out the computation, note that the prescription for Noether™s theo-

rem given in (1.13, 1.14, 1.15) is particularly simple for the case of the functional

Λ = ω1 § · · · § ωn

on the contact manifold M 2n+1. This is because ¬rst, dΛ = Π already, so no

correction term is needed, and second, the in¬nitesimal Euclidean symmetries

(prolonged to act on M ) actually preserve Λ, and not merely the equivalence

class [Λ]. Consequently, the Noether prescription is (with a sign change)

·(v) = v Λ.

This v Λ is an (n ’ 1)-form on M which is closed modulo the Monge-Ampere

system EΛ .

Proceeding, we can suppose that our translation vector ¬eld is written up

on the Euclidean frame bundle as

vF = Ae0 + Ai ei ,

where the coe¬cients are such that the equation dv = 0 holds; that is, the

functions A and Ai are the coe¬cients of a ¬xed vector with respect to a varying

oriented orthonormal frame. We easily ¬nd

n

Ai ω(i).

•v = v F Λ=

i=1

This, then, is the formula for an (n ’ 1)-form on F which is well-de¬ned on the

contact manifold M and is closed when restricted to integral manifolds of the

Monge-Ampere system EΛ. To see it in another form, observe that if we restrict

our (n ’ 1)-form to a transverse Legendre submanifold N ,

Ai (—ωi ) = — v, dx .

• v |N =

Here and throughout, the star operator — = —N is de¬ned with respect to the

induced metric and orientation on N , and the last equality follows from the

equation of En+1 -valued 1-forms dx = e0 θ + ei ωi , where θ|N = 0. We now

have a linear map from Rn+1, regarded as the space of translation vectors v, to

the space of closed (n’1)-forms on any minimal hypersurface N . Tautologically,

such a map may be regarded as one closed (Rn+1)— -valued (n ’ 1)-form on N .

Using the metric to identify (Rn+1)— ∼ Rn+1, this may be written as

=

•trans = —dx.

This is the meaning of the conservation law stated at the beginning of this

section. Note that each component d(—dxa) = 0 of this conservation law is

1.4. HYPERSURFACES IN EUCLIDEAN SPACE 29

equivalent to the claim that the coordinate function xa of the immersion x :

N ’ En+1 is a harmonic function with respect to the induced metric on N .

Turning to the rotation vector ¬elds, we ¬rst write such a vector ¬eld on

n+1

E as

n+1

‚

xa R b b , Ra + Rb = 0.

v= a b a

‚x

a,b=1

It is not hard to verify that this vector ¬eld lifts naturally to the frame bundle

F as

‚ ‚

xa R b A b c + Ab Rb Aa c ,

vF = ac cad

‚ω ‚ωd

‚ ‚

where the coe¬cients Aa are de¬ned by the equation ‚ωa = Ab ‚xb , and the

a

b

‚ ‚

tangent vectors ‚ωa , ‚ωa are dual to the canonical coframing ω a , ωb of F.

a

b

We can now compute (restricted to N , for convenience)

xa Rb Ab ω(i)

(vF Λ)|N = ai

—(xa Rb Ab ωc )

= ac

— R · x, dx .

=

Reformulating the Noether map in a manner analogous to that used previously,

Rn+1 ∼ so(n + 1, R)— -valued (n ’ 1)-form on N

2

we can de¬ne a =

•rot = —(x § dx).

Once again, •rot is a conservation law by virtue of the fact that it is closed if

N is a minimal hypersurface.

It is interesting to note that the conservation law for rotation symmetry is

a consequence of that for translation symmetry. This is because we have from

d(—dx) = 0 that

d(x § —dx) = dx § —dx = 0.

The last equation holds because the exterior multiplication § refers to the En+1

where the forms take values, not the exterior algebra in which their components

live. It is an exercise to show that these translation conservation laws are

equivalent to minimality of N .

Another worthwhile exercise is to show that all of the classical conservation

laws for the H = 0 system arise from in¬nitesimal Euclidean symmetries. In the

next chapter, we will see directly that the group of symmetries of the Poincar´-e

Cartan form for this system equals the group of Euclidean motions, giving a

more illuminating proof of this fact. At the end of this section, we will consider

a dilation vector ¬eld which preserves the minimal surface system E, but not

the Poincar´-Cartan form, and use it to compute an “almost-conservation law”.

e

By contrast, in this case there is no discrepancy between g[Λ] and gΠ. To

see this, ¬rst note that by Noether™s theorem 1.3, gΠ is identi¬ed with H n (EΛ ),

´

30 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

and g[Λ] ⊆ gΠ is identi¬ed with the image of the connecting map δ in the long

exact sequence

δ ι

· · · ’ H n’1(„¦— /EΛ) ’ H n(EΛ ) ’ HdR (M ) ’ · · · .

n

With M ∼ En+1 — S n , we have the isomorphism HdR (M ) ∼ R obtained by

n

= =

n+1

integrating an n-form along a ¬ber of M ’ E , and it is not hard to see that

any n-form in EΛ must vanish when restricted to such a ¬ber. Therefore the

map ι is identically 0, so δ is onto, and that proves our claim.

Interpreting the Conservation Laws for H = 0

To understand the meaning of the conservation law •trans, we convert the equa-

tion d•trans|N = 0 to integral form. For a smoothly bounded, oriented neigh-

borhood U ‚ N ‚ En+1 with N minimal, we have by Stokes™ theorem

—dx = 0.

‚U

To interpret this condition on U , we take an oriented orthonormal frame ¬eld

(e0 , . . . , en ) along U ∪ ‚U , such that along the boundary ‚U the following hold:

±

e0 is the oriented normal to N,

en is the outward normal to ‚U in N, (1.20)

e1 , . . . , en’1 are tangent to ‚U.

Calculations will be much easier in this adapted frame ¬eld. The dual coframe

ωa for En+1 along U ∪ ‚U satis¬es

n’1

0

ei ω i + e n ω n .

dx = e0 ω +

i=1

Now, the ¬rst term vanishes when restricted to N . The last term vanishes when

restricted to ‚U , but cannot be discarded because it will a¬ect —N dx, which we

are trying to compute. Consequently,

n’1

—N dx = ei ω(i) + en ω(n).

i=1

Now we restrict to ‚U , and ¬nd

(’1)n’1en ω1 § · · · § ωn’1

—N dx|‚U =

(’1)n’1n dσ.

=

Here we use n to denote the normal to ‚U in N and dσ to denote the area

measure induced on ‚U . Our conservation law therefore reads

n dσ = 0.

‚U

1.4. HYPERSURFACES IN EUCLIDEAN SPACE 31

In other words, in a minimal hypersurface the average of the exterior unit normal

vectors over the smooth boundary of any oriented neighborhood must vanish.

One consequence of this is that a minimal surface can never be locally convex;

that is, a neighborhood of a point can never lie on one side of the tangent plane

at that point. This is intuitively reasonable from the notion of minimality.

Similar calculations give an analogous formulation for the rotation conservation

law:

(x § n)dσ = 0.

‚U

These interpretations have relevance to the classical Plateau problem, which

asks whether a given simple closed curve γ in E3 bounds a minimal surface.

The answer to this is a¬rmative, with the caveat that such a surface is not

necessarily unique and may not be smooth at the boundary. A more well-posed

version gives not only a simple closed curve γ ‚ E3 , but a strip, which is a curve

γ (1) ‚ M consisting of a base curve γ ‚ E3 along with a ¬eld of tangent planes

along γ containing the tangent lines to γ. Such a strip is the same as a curve

in M along which the contact 1-form vanishes. Asking for a minimal surface

whose boundary and boundary-tangent planes are described by a given γ (1) is

the same as asking for a transverse integral manifold of EΛ having boundary

γ (1) ‚ M .

The use of our two conservation laws in this context comes from the fact that

(1)

γ determines the vector-valued form —N dx along ‚N for any possible solution

to this initial value problem. The conservation laws give integral constraints,

often called moment conditions, on the values of —N dx, and hence constrain the

possible strips γ (1) for which our problem has an a¬rmative answer. However,

the moment conditions on a strip γ (1) are not su¬cient for there to exist a

minimal surface with that boundary data. We will discuss additional constraints

which have the feel of “hidden conservation laws” after a digression on similar

moment conditions that arise for boundaries of holomorphic curves.

It is natural to ask whether a given real, simple, closed curve γC in complex

space Cn (always n ≥ 2) is the boundary of some holomorphic disc. There is a

di¬erential ideal J ‚ „¦— (Cn ) whose integral manifolds are precisely holomor-

R

phic curves, de¬ned by