Math. Ann. 289, 341-354 (1991)

Amalm �9 Springer-Verlag 1991

The cone over the Clifford torus in R 4 is R-minimizing

Frank Morgan

Department of Mathematics, Williams College, WiUiamstown, MA 01267, USA

Received August 6, 1990; in revised form September 14, 1990

1 Introduction

This paper shows as announced in [M4] that the regularity results for area- minimizing hypersurfaces (integral currents) do not hold for hypersurfaces minimizing the integrals of all other elliptic integrands ~. In particular, we give a singular 3-dimensional O-minimizing hypersurface in R 4. This example shows the sharpness of the results of Almgren, Schoen, and Simon [AlmSS, Theorem II.7], which guarantee regularity up through R 3.

The surface is the cone C over the Clifford torus S 1 x S 1CR2x R2:

C= {(x, y)e R 2 x R 2 : Ixl=lyl<l}.

The integrand �9 is position-independent ("constant-coefficient"), and as a function on unit normals depends smoothly on 0 = tan-I (lYl/Ixl) alone. The unit O-ball is pictured in Fig. 1.0.1. Any smooth, symmetric, uniformly convex approximation of the square will do. Note that �9 is smaller (say 1) on the normals to diagonal directions, which occur in the cone C, so that ~ O(n) is relatively small.

c The unit ball of the norm dual to ~, called the Wulffcrystal W(O), is pictured in

Fig. 1.0.2. The Wulff crystal itself solves an important problem: its boundary surface S minimizes O(S) for fixed volume enclosed (cf. I-T, Sect. 1]). In nature O(S) represents the surface energy of a crystal, and the Wulff crystal W(O) gives the shape which a fixed volume of material assumes to minimize surface energy. The Wulff crystal of our norm �9 resembles a pivalic acid crystal, see Fig. 1.0.3.

Hypersurfaces which minimize area are regular up through R 7 (IS, Theorem 6.2.1]; see [M3, Sect. 8.2, Sect. 10], [F1, Theorem 5.4.15]). Similarly, hypersurfaces minimizing for small smooth perturbations �9 of the area integrand are regular up t h rough R 7, too [AlmSS, Theorem 11.8]. Before this paper, there were no examples of singularities below R a for any elliptic integrand.

The proof is the first specific application of calibrations to integrands other than area.

342 F. Morgan

1 0=-1

Fig. 1.0.1. The unit ~-ball. The smallness of �9 in the diagonal directions helps to make the cone C ~-minimizing

Fig. 1.0.2. The Wulff crystal W(~) may be defined as the unit ball for the norm dual to ~. For fixed volume, W(r has the least surface energy measured by r

1.1 Elliptic integrands. The energy of a soap film is almost exactly proportional to its area. The surface energy of a salt crystal, however, depends on the direction, too: certain directions expose more bonds than others, see Fig. 1.1.1.

For oriented surfaces S of dimension n - 1 in R ~, the energy ~(S) is determined by an integrand ~: S ~ - 1 ~ R :

@(S) = S r S

where n is the odentexi unit normal to S. One expects good regularity results for e-minimizing surfaces if ~ is "elliptic," i.e. the unit e-ball is uniformly convex, which guarantees that flat hyperplanes are uniquely e-minimizing. The basic definitions and results are given in Sect. 3.

A number of materials have elliptic surface energies and smooth, uniformly convex crystals. For helium, the energy is almost isotropic and the crystal is nearly spherical. Recent experiments of Glicksman and Singh [GS] reveal a very anisotropic elliptic energy for pivalic acid (PVA) and provide nice photographs, see Fig. 1.0.3.

1.2 Proof by calibration. The proof that the cone C over S 1 x S I c R 4 is e-minimizing employs the "method of calibrations" (cf. [HL, Introduction]). One must produce a closed differential 3-form or "calibration" ~p such that for any point p and unit 3-plane ~, with unit normal ,~,

(~, ~0(p)) < ~(,~), (1)

The cone over the Clifford torus in R 4 is 4~-minimizing 343

Fig. 1.0.3. A pivalic acid crystal resembles the Wulff crystal W(q0 of 4~ [GS]

Fig. 1.1.1. Soap bubbles and salt crystals minimize different kinds of surface energies

with equali ty whenever ~ is the oriented unit tangent to C at p. Then if S is any other surface with the same boundary ,

�9 (C) = I ~o = ~ ~o __< ,P(S), C S

so that C is g-minimizing. Finding a cal ibrat ion r remains an art, not a science. O u r cal ibrat ion

r = (sin 220dr + sin 40rdO)/` ]x]dO~/` ly]d02,

where rZ=lxlZ+ly] 2, O=tan-l(lyl/Ixl), 0 1 = a r g x , Oz=argy. I t resembles the 7-form of Federer ' s p roof IF, Sect. 6.3] after Lawson [L, Sect. 5] tha t the cone over S 3 x S 3 is area-minimizing. At any point in our cone C, 0 = n/4, and q~(n/4) = d r / , Ixld01 ^ ]yld02 is precisely dual to each unit tangent ~o to C. Hence

<~, ~o(p)> < 1 < q~(,~),

with equali ty whenever ~ = ~o. Thus (1) holds at points p e C. Unfor tunate ly , for p r C (for example 0 = n/8), the sin40 term, which is necessary to m a k e ~0 closed, tends to m a k e q~ big. In order for (1) to hold, the largeness of ~o(n/8) must be somehow compensa ted for by the largeness of ~b(n/8).

Establ ishing the est imate (1) at all points a lmost a lways is a main difficulty. F o r the case of area, the r ight -hand side is 1, and the es t imate becomes

k0(p)l < 1, independent of ~. Fo r a general in tegrand 4, the es t imate involves bo th p

344 F. Morgan

and ~. This difficulty explains why calibrations have not been applied specifically to integrands other than area before.

We handle this difficulty with a lemma that associates with rp the function on unit vectors

G(w)=sup{Iq~(p)l: w is the oriented unit normal to the (n-1)-plane dual to ~p(p)}.

The lemma says that the desired estimate (1) holds if the graph of G lies inside the Wulff crystal W(O), thus reducing the required estimate to a single parameter.

2 Easy examples o f singular ~-minimizing hypersurfaces

This section continues the introduction with examples of singularities in hypersurfaces in R n minimizing the integral of an integrand O. For these examples, either the ambient dimension n_>_ 8 or the integrand �9 is not elliptic. Definitions and the main body of the paper begin in Sect. 3.

The cone C over S 3 x S 3 in R s was the first known example of a singular area- minimizing hypersurface [BDG, 1969]. To obtain trivially other examples, let f : R S ~ R 8 be a diffeomorphism. If C is O-minimizing, then f(C) is f(O)- minimizing.

Actually, for many perturbations �9 of the area-integrand Oo, the cone C over S 3 x S 3 is O-minimizing. More specifically, if �9 is any constant-coefficient integrand such that

�9 (~) > Oo(0,

with equality whenever r is an oriented unit tangent plane to C, then C is O-minimizing. Indeed, let C' be any other surface with the same boundary. Then

�9 (c') __> Oo(C')_-> Oo(C') = o ( c ) .

Therefore, C is O-minimizing. This same argument shows that complex-analytic varieties are O-minimizing

for many smooth elliptic integrands O. If �9 is not required to be elliptic, it is easy to obtain singular O-minimizers, even

for curves in the plane. For example, consider the borderline nonelliptic integrand pictured in Fig. 6.2.1 (of. Sect. 6.2). All curves C from Pl = ( - 1, 0) to P2 = (1,0) in R 2 with angles 0 with the positive x-axis satisfying 101 < ~/4 have the same value for ~C). Hence the singular sets of O-minimizers are essentially arbitrary. For example, one can construct such a O-minimizing curve bounded by 2p2-2pl with a Cantor-like set of singularities of arbitrary dimension between 0 and 1, see Fig. 2.1 (of. [M3, Sect. 3.12]). Such constructions easily generalize to higher dimensions.

. . . . . . ~ . . . . . . . . ,r . . . . . . . . v

(-1 ,o) (1,0)

Fig. 2.1. This r curve has a Cantor-like singular set of dimension 1/2

The cone over the Clifford torus in R 4 is ~-minimizing 345

3 EIHptic integrands

Here begins the main body of this paper. We consider only hypersurfaces in R" and positive constant-coefficient integrands of degree n - 1 . In these cases the definitions of integrand and ellipticity can take the simple form given in 3.1 below.

Roughly, an integrand �9 assigns to each unit normal direction v a cost or energy ~(v). If S is a hypersurface, define

�9 (S)= s

where n is the unit normal to S at x. For example, if ~/i is the area integrand defined by q/io(v ) = 1, then ~/io(S ) = area S.

For our hypersurfaces we use the oriented, possibly singular rectifiable currents of geometric measure theory. See [M2, Sect. 1.4, Sect. 4.2] or [F1, Sect. 4.1.24] for basic definitions and terminology.

3.1 Definitions. A (constant-coefficient, positive, parametric) integrand is a positive, continuous function � 9 S"- 1 --.R. Occasionally, it is convenient to view as a homogeneous function on R" -{0} , so that ~(v)= Iol~(v/Ivl).

Of course, an integrand ~ is convex if ~(vl + Vz) ~ ~(vl) + ~(v2). An integrand is elliptic if the integrand F(v)=~(v)-clv] is convex for some c>0 , or

equivalently, if the unit ~/i-ball

{0eR": I} is uniformly convex. If r is C 2 on S"- i, then uniform convexity is equivalent to positive inward curvature at every point in every direction. Note that the unit Q-ball is the region bounded by the radial plot of I / ~ : S " - I ~ R +. The general definition of ellipticity in [F I, 5.1.2] is equivalent to our definitions for integrands of degree n - 1.

An integrand ~ is determined by its radial plot

P= {rveR": y e s "-1, r = ~(v)}.

By a unit normal n to P at v e S"- ~ we will mean any unit vector such that n . p < n . p o for all peP, where po=~r~(v)v, see Fig. 3.1.1.

n

Fig. 3.1.1. A unit normal n to P at v. In this case there is a whole interval of unit normals to P at v

346 F. Morgan

If P is strictly convex, there is at least one unit normal at each v ~ S"- 1 and v is a function of n: v =f(a). f is Lipschitz if and only if P is uniformly convex.

On the space of integrands, we follow [Fu] and I-T, p. 573] in defining operators W,A, I by

W(~)(w) = inf {(v. w)- 1 ~(v): v ~ S"- 1, v- w > 0},

A(~)(v) = sup {(v" w)~(w): w e S"- 1 },

l(~)(v) = l/~(v),

for y e S *-1.

Remark. W is the famous Wulff operator. The radial plot B of W(~) bounds the Wulff crystal of ~, which for fixed volume minimizes ~ (B) - = ~ ~. Therefore,

B

physical crystals which minimize surface energy given by �9 assume that shape. The classical Wulff construction is essentially the fact that the Wulff crystal is the intersection of all the halfspaces

Ho= {w: w. v<<_~v)} (cf. Fig. 3.3.1).

The following consequences of the definitions are not hard to verify. Property (3) is essentially equivalent to the statement that the double dual of a norm on a finite-dimensional vectorspace is the norm itself (cf. Remark 3.4).

3.2 Proposition. The operators W, A, 1 have the following properties. (1) W = I o a o I , A = I o WoL (2) I f ~ 1 < ~ 2 , then W(+t)< W(+2) and A(+0__<A(+2). (3) / f the radial plot of G is convex, then Wo A(G) = G. Likewise, if ~ is convex,

then A o W(+)= ~.

The following proposition should be standard, although I have not found it in the literature (of. JR, Theorem 26.3]).

3.3 Proposition. Let tb be a convex integrand, with Wulff functional W(+). (1) ~ is C 1" 1 if and only if W(~) has a uniformly convex radial plot. (2) �9 is elliptic if and only if W(~) is C 1' 1. (3) / f W(~) has a uniformly convex radial plot and W(~) is C k [real-analytic],

then �9 is C ~- 1 [real-analytic]. (4) I f �9 is elliptic and C ~ [real-analytic], then W(~) is C k- 1 [real-analytic].

Proof. Let G = W(~). By 3.2(3), �9 = A(G). First suppose that G(W) has a uniformly convex radial ptot.-Then givett any unit norrnat-wto the radial plot, w is determined as a Lipschitz function of v: w =f(v). The supremum in the definition of A(G)(v) is attained at w, see Fig. 3.3.1. Comparison of the two definitions shows that the supremum in the definition of A(1/~)(w) is attained at v. Thus w =f(v) is the unit normal to the radial plot of 1/~ at v. Consequently, 1/~ and hence ~ are C 1' 1

The same argument run backwards proves the converse and establishes (1). Moreover, if G = W(~) is C k [real-analytic], then w =f(v) is C k- 1 [real-analytic] and

~v) = a(G)(v) = (v. w)G(w)

is C k-x [real-analytic] proving (3). To prove (2) and (4), apply (1) and (3) to ~ ' = I o W(+) and W(+')=I(+).

The cone over the Clifford torus in R 4 is ~-minimizing 347

Fig. 3.3.1. A smooth elliptic integrand r and its Wulff crystal G= W(~). r =(v. w)C_,(w)

Fig. 3.4.1. A pair of dual unit norm balls. One is smooth but not uniformly convex. The other is uniformly convex but not smooth

3.4 Remark. A symmetr ic convex integrand �9 defines a norm. The dual n o r m is given by I o W(~). F r o m this perspective, Propos i t ion 3.3 says that a unit no rm ball is uniformly convex if and only if the dual no rm ball is C 1' 1. Figure 3.4.1 gives two dual unit n o r m balls, one of which is smoo th but no t uniformly convex, the other of which is uniformly convex but not smooth.

The distinction between C1' 1 and differentiable bears on applications as well as

theory. F o r p > 2, 1_ + 1 = 1, 1 < q < 2, the L p and L ~ Holder no rms are dual on say P q

348 F. Morgan

Fig. 3.4.2. Similarly, the unit L p ball is C x' ~ but not uniformly convex at 4 points, while the unit L ~ ball is uniformly convex but not C ~' ~ at 4 points

R 2. Their unit balls are both differentiable and strictly convex. However, the unit L p ball is not uniformly convex, and the unit L ~ ball is not C 1' 1, see Fig. 3.4.2.

4 Calibrations

The method of calibrations (cf. [H; HL; M 1; M2]), which provides many examples of rectifiable currents T in R" which minimize the integral of the area integrand ~o, extends directly to other integrands ~. Proposition 4.1 gives a special case of results of Federer [F2, 4.10(5), 6.2]. Calibrations for more general integrands are studied by Dao I-D], too.

4.1 Proposition. Let ~:S" -* ~ R be an integrand on R". Let E be a closed set with ~ " - * ( E ) = O. Let tp be a C 1 closed differential ( n - 1)-form on R " - E . Suppose that for all z e R " - E, unit (n -1)-planes 4,

<~, ~(z)>__< ~(.~). (I)

Let T be a rectifiable current such that for II T[l-almost all z,

< ~(z), ~(z)> = ~(. ~(z)).

Then T is r

Proof. As pointed out in [HL, Lemma 4.8], it follows from a general result on removable singularities [HP, Theorem 4.1b] that if dcp=0 off a set of A "~"-* measure 0, then d~p = 0 weakly on all of R". Hence if S is another rectifiable current with dS=~T, then S(q~)= T(cp). Now

~ T ) = ~ ~(*T)dlt TII = ~ < ~(z), q,(zDdll TII

= T(~p) = S(~p) = ~ <~(z), ~p(z)>dllSl[

=< f ~(*g)dllS[I = MS).

Therefore, T is ~-minimizing. Hypothesis (1) of Proposition 4.1 can seem quite difficult to obtain for a general

integrand ~ and variable-coefficient form ~p. The following novel lemma obtains such an estimate by relating an integrand associated with ~p to the Wulff crystal of r

The cone over the Clifford torus in R '~ is ~-minimizing 349

4.2 1.emma. Let �9 be an integrand on U C R". Let r be a differential ( n - 1)-form. Let G be the associated integrand

a(w) = sup {l~p(z)l: w is the oriented unit normal to the (n-1)-plane cp(z)*}.

Then

(~, ~o(z)) <= ~(.~),

for all z e U and unit (n-1)-planes ~, if and only if G< W(ep).

Remarks. Here the Hodge dual *~ of r is just the oriented unit normal to ~. For the purposes of this lemma, it is convenient to define sup~b=0 and not

insist that the integrand G be strictly positive or continuous.

Proof. Let w denote the oriented unit normal to ~p(z)*. Then

Consequently, <r cp(z)> = (* r w)I~p(z)l.

<r ~(z)> __< ~(.~)

~" [~p(z)l < (*~. w)- x~(.~) whenever *~- w > 0

~. 6__< w(~)

by the definition of W(~) (3.1).

5 Polar coordinates and the reduction to R 2

This paper proves the cone over S t x SXcR 4 to be R-minimizing for an appropriate integrand # by producing a suitable calibration cp. By taking # and cp, as well as S 1 x S 1, to be SO 2 x S02-invariant, the analysis essentially reduces to R 2. This section explains that reduction.

5.1 Polar coordinates r, 0 on R". On {(x, y)~ R"' x R "2} = R", we will employ the lYl "polar coordinates" r = i/Ixl 2 + lyl 2 and 0 = t an - t ~-/e [0, ]]. Bars will denote the

same coordinates on the tangent space. The subscript 0 will be used to distinguish the case of R x R = R 2.

If # : [0 , -~]~R, we will sometimes also denote by �9 the integrand ~ o 0 : S . - I ~ R .

5.2 Proposition. Let ~ be a positive continuous function on [0,~]. I f ~(~0) is an elliptic integrand on R x R, then ~(ff) is an elliptic integrand on

R "1 x R "2 for any positive integers hi, n2. I f r is C OO [real-analytic] on R x R, then ~ 0 ) is Coo [real-analytic] on R "~ x R "~.

We omit the proof.

5.3 Proposition. Let ~ be a positive continuous function on [0,~], so that 0(0), 0(0o) give integrands on R "' x R"2=R" and R 1 x R 1 = R 2. Let

q,(O)= p(O)dr + q(O)rdO ,

CPo(0o) = P(Oo)dro + q(Oo)rodOo

give 1-forms on (R " ' - {0}) • (R "2 - {0}) and ( R - {0}) • ( R - {0}).

350 F. Morgan

I f for all unit vectors v in R 2, at all points in ( R - {0}) x ( R - {0}),

(v, ~o(0o)) _-< ~(.v) , (1) then for all unit ( n - 1)-planes ~ in R n, at all points in (R n ' - {0}) x (R n2- {0}),

( ~, ~(0) ^ Ol(X) ^ O2(y)) < #(*~), (2)

where 01, f2 z denote the normalized solid angle forms on R"' - {0}, R "2 - {0}: x

t21(x)= -~[ j (dx I ^ ... ^ dx,,),

I22(x)= ]-~[j (dyl ^ ... ^ dye2).

Moreover, equality holds in (2) if and only if ~kt21 ̂ ~'~2 is a unit vector in

{ 0 , f__O} =__R2 for which equality holds in (1)" span ~r r

We omit the proof.

6 The cone over S t • S t is r

This section proves our main theorem (6.2), that the cone C over S~x S t is R-minimizing. The proof defines both the elliptic integrand �9 and an associated calibration r on R 4. The cone C is "calibrated" by ~ and therefore R-minimizing.

First, we need one lemma which later yields the essential estimate on the calibration r

6.1 Lemma. For q~o=sin2Oo(sin2Oodr+ 2cos2OordOo), consider the associated integrand G: S 1 ~ R

G(w)=sup{[q%(Oo)l: w is the oriented normal to q~o(0o)*}.

By symmetry, G(w) = G(~o) is a function of U o alone. Then G(Uo) < sec(~ /4- 0-o), with equality if and only if t7 o = ~/4. Moreover, at ~o = ~/4, G is uniformly convex.

We omit the proof, mainly some computations, see Fig. 6.1.1. The following theorem is the main result of this paper.

00)

Fig.6.1.L G(O'o) lies inside the diamond see - g o , with contact at the four points where

~7o = ~/4

The cone over the Clifford torus in R 4 is O-minimizing 351

6.2 Theorem. There is a real-analytic elliptic integrand ~:S3--~R 4 such that the cone over S 1 x S 1CR 4 is R-minimizing,

Proof. Consider the integrand on R 2

~P(0-o) = V~ max {cos 0-o, sin 0-o } (~o ~ [0,~])

(see Fig. 6.2.1). It is easy to check that its Wulff crystal (cf. 3.1) is the d iamond / \

W(~)(Oo)=sec( : - - - t7o) . Unfortunately, this crystal is neither smooth nor \ - - /

uniformly convex. Consider also the 1-form

tpo = sin20o[sin 20odr + 2cos20ord0o] (1)

on ( R - {0}) x ( R - {0}). Let G be the associated integrand G(w) = sup {[~po(0o)l: w is the oriented unit normal to ~(0o)*}. By Lemma 6.1, ~0-o)<W(~)(~o), with

rc if and only if 0o = ~- (see Fig. 6.2.2). Moreover, at ~-o = ~ , G is uniformly equality

convex. Therefore, there is a real-analytic, uniformly convex integrand H,

Fig. 6.2.1. The unit ball for the borderline nonelliptic integrand ~v and its Wulff crystal W(~v). Our desired elliptic integrand @ will be a perturbation of ~v

H

Fig. 6.2.2. Since G_~W(~), with equality and uniform convexity at ~/4, there is an H, G < H < W(~'), uniformly convex everywhere

352 F. Morgan

G < H < W(~W). For example, one could take for N large and e small,

F(coS0-o + sinO-o~ 2" + (c~ singo~2"] -'/2N H(•o)=(1-e)[ \ - ~ } \ ~ ] j +e.

Of course, there arc many other possibilities. Let r =A(H), so that H= W(r see Fig. 6.2.2. Then G(Bo)=< W(O)(0o), with

equality if 0o = ~/4. Now by Lcmma 4.2, for all unit vectors v in R 2, at all points in (a- {o}) • (n- {o}),

(v, goo(0o))-<_ (2) Moreover, we claim that equality holds if 0o =zc/4 and v=~/~r. Indeed,

(v, goo(0o)) = (~/Or, dr) = 1 = P(*v), because go(*V) = ~ - go(V) = { - 0o = ~z/4 and ~v(zc/4) = 1. Since ~'__< r equality must hold as claimed.

Now we consider R 4 = R 2 x R 2. Since O(go) is a real-analytic, elliptic integrand on R 2, by Proposition 5.2, r is a real-analytic, elliptic integrand on R 4. We will show that the cone over S* x S 1 is e-minimizing by calibrating it (Proposition 4.1) by the 3-form on (R 2 - {0}) x (R 2 - {0})

go = goO(0) A ~'~I(X) A ~"~2(Y),

where goo was defined above (1), and f~l, f2z denote the normalized solid angle forms defined in Proposition 5.3.

To complete the proof, we must show that go is dosed and that for all (x , y) e ( R 2 - {0}) x (a 2 - {0}), for all unit ( n - 1)-planes 4,

go(x)) _<__ (3)

with equality when ~ is the oriented unit tangent to the cone over S 1 x S ~ at (x, y). To prove that go is closed, we note that for

u(x, y) = ~r 3 sin320,

du = �89 2 sin 20(sin 20dr + 2cos 20rdO),

go is the wedge product of du with the (unnormalized) closed solid angle forms

"dO{" = ~l/ lxl ,

"dO 2" = f&/lYl .

Inequality (3) follows from inequality (2) by Proposition 5.3. Moreover, equality holds if 0o=~C/4 and V=~LI21At22=3/Or. We must show equality holds for almost every unit tangent plane ~ to the cone C over S t x S 1.

So let ~ be a unit tangent plane to C. Firstly, for every (x, y)e C-{0} , Oo(x, y) = g/4. Secondly, ~ is the wedge product of the radial direction a/ar with the two solid angle directions t2* and f2*:

~=~r A ~'~A ~-~.

Hence v = ~ L fl l A f12 = Orr as desired.

Therefore, go calibrates the cone C over S 1 x S ~ as Proposition 4A describes, and C is O-minimizing.

The cone over the Clifford torus in R 4 is ~-minimizing 353

6.3 Remarks. The theorem and proof hold with the same integrand ~0) ,

0-= t a n - ~ --[Yl, for the cone over S ~ x S ~ in R 2~ + 2 for any x > 1. To produce a closed

~p, one takes

1 u = 1) r2~+ x sinP20,

2~(2x +

~p=du A t21A ~ 2

=sinP-l-~20(sin2Odr+2~+lCOS2OrdO ) A ['21 ̂ f l 2 .

The condition that G be uniformly convex at x/4 is equivalent to p >

A convenient choice is p = 2x + 1. Then

~o = sin~ 20(sin2Odr + 2cos2OrdO) A ~1 A t22

8x 2 + 2 r - 1

8 x - - 4

and the estimates are clearly easier for r > 1 than for x = 1. As the p roof shows, the integrand is by no means unique. The cone over S ~ x S ~

(x > 1) is ~-minimizing for an infinite-dimensional family of elliptic integrands ~. The cone over S ~ S o in R 2 is no t minimizing for any elliptic integrand,

because straight lines are uniquely minimizing for any elliptic integrand IF1, 5.1.2]. However, it is minimizing for the convex integrand T of Fig. 6.2.1, t hough not uniquely.

Acknowledgements. I would like to thank Williams students Ken Hedges, Rajiv Kochar, Lisa Kuklinski, Adam Levy, Zia Mahmood, and Kob Pootrakool for their help in this research. An improvement from R 6 to R 4 was obtained while I was visiting J. M. Coron at the Universit6 de Paris-Sud, Centre d'Orsay, in June, 1988.

This work was partially supported by a grant from the National Science Foundation.

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354 F. Morgan

[M1]

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[M4]

[R] [S] C~

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