RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo XXXIII (1984), pp. 37-46

AREA-MINIMIZING CURRENTS BOUNDED BY HIGHER MULTIPLES OF CURVES

FRANK MORGAN

For any positive integer n, we give an example of a smooth Jordan curve C in IR 4 such that the smallest integral current bounded by nC has less than n/k times the area of smallest integral current bounded by kC ( lgk<n) .

1. Introduction.

L. C. Young [8] gives an example of a curve Co in IR ~ such that the smallest integral current (oriented surface) bounded by Co has more than half the area of the smallest integral current bounded by 2 Co:

I (Co) > (1/2) I (2 Co).

Can ( l / k ) I (k C) continue to decrease as k increases? Of course, if N (C) denotes the area of the smallest normal current (generalized surface with real rather than integer multiplicities) bounded by a rectifiable curve C, then

( ~ Ik) t (k/C) >_ N (c),

for all integers k. Herbert Fcderer [2,5.8] proves that

(1) lim ( l /k) I (k C)----N (C).

Young's construction exploits the nonorientability of the Klein bottle; it seems

3 8 FRANK MORGAN

that

(2) (1/2) I (2 Co) = N (Co),

and he suggests that (2) might hold for all rectifiable curves [8, p. 259].

Brian White, in 1981, and the author, early in 1982, independently esta- blished, for any integer n, different examples of a curve C such that

(3) (1/k) I (k C) > N (C) (I <_ k < n).

For the author's example, it is proved that

(1 In) I (n C) = N (C).

White generalizes Young's example. Our example exploits symmetry. Although the symmetric curve C we present has n + 1 components, any smooth Jordan curve which closely approximates C in the flat norm satisfies (3) as well.

Assuming a suitable bridge theorem, one infers the existence of a piecewise smooth Jordan curve C' in IR 4 such that

( l /k) I (k C')>N (C') k = 1 , 2 , 3 ....

The proo/ 1.1. In our proof we show that any area-minimizing normal current P bounded by our curve C is supported on a certain oriented mani- fold M, with O M = n C (as currents). We deduce that P=(1/n)M. Therefore the unique area-minimizing normal current bounded by kC is kP=(k /n )M, which is integral if and only if k is a multiple of n.

To show that P is supported on M, we make use of the symmetry of C under a circle of isometries F. We can assume that P is F-invariant, since averaging P over F can only increase its support. Modding out by the action of F, we reduce the problem to identifying the one-dimensional normal current, invariant under 120 ~ rotations about an axis in IR 3, bounded by 2n given points, which minimizes length in a certain metric. See Figure 1.1(1).

The unique solution consists of n 2 oriented geodesic segments, each with multiplicity 1/n. M is their orbit under F, and P is supported on M.

We need to prove a lemma (2.2), which implies that the support of a minimizing normal current bounded by a finite set of points is contained in the closure of the union of the supports of all minimizing integral currents with the same boundary. This lemma holds for integrands more general than 8tea,

AREA MINIMIZING CURRENTS BOUNDED, ETC. 39

p"

120"

qo

_ • / GO G I G 2

P

Figure I.I ( I )

Cose n = 3 , schernot ic

p '

Remarks 1.2. One can obtain similar examples in higher dimensions by taking the Cartesian product of our examples with oriented intervals. However, if C is a 0-dimensional or (N--2)-dimensional integral current boundary in IR N, then

I ( C ) = N ( C )

[2,5.I0,5.I2]. Relation 1 (I) holds in all dimensions and codimensions. Federer [2] studies a more general question about convex parametric

integrands on manifolds. F. J. Almgren, Jr., has pointed out that the examples of Young, White,

and the author lead to counterexamples to the statement that the Cartesian product of two area-minimizing integral currents in IR ~ be area-minimizing. For any integer n>_2, let S be an area-minimizing integral current such that n S is not area-minimizing. Then the Cartesian product of S and a suitably long interval of multiplicity n is not area-minimizing. The corresponding ques- tions for normal currents and flat chains modulo ,~ remain open.

This work was supported in part by the National Science Foundation.

40 FRANK MORGAN

2. Preliminaries.

We refer to Federer's treatise [1] for the basic definitions, notations, and results about spaces of currents, boundat 3, a, mass M, integrands O, and ~- minimizing currents with respect to a pair (A, B). We begin by recording as Lemma 2.1 a special case of a result of Federer. The lemma says that a one- dimensional (I)-minimizing rectifiable current is still minimizing in the larger class of flat chains. Lemma 2.2 goes on to show that the support of any one- dimensional O-minimizing flat chain is contained in the closure of the union of the supports of the tl)-minimizing rectifiable currents.

LEMMA 2.1 [2,5.12]. Let Z be an open subset o] IR N, B c A ~ Z compact.

Let �9 be a positive, convex, even parametric integrand on Z. Let C E ~o.a (Z). Then

inf {O (S): S(~ ~,.a (Z), spt (a S - - C ) c B } =

=inf { �9 (Q): Q E Ft,a (Z), spt (0 Q - C) c B }.

LEMMA 2.2. Let Z be an open subset of IRN; let B o A be compact Lipschitz neighborhood retracts in Z. Let r be a positive, convex, even parametric inte- grand on Z. Let OoE F~,A (Z) be absolutely O-minimizing with respect to (A, B).

Suppose there is an SoVz ~,A (Z) with spt (O So--a Oo)cB. Let

E = U {sptS: S E~ .a (Z) , S is O-minimizing with

respect to (A, B), spt (0 S-- 0 Oo) c B }.

Then spt OocE.

Proof. Suppose sptQoCE. Ook__B=O, since otherwise consideration of Qo[_.(Z--B) would contradict the minimizing property of Q0. Therefore

Qo[__(Z--E--B)#O. Choose a C ~ function q~ on Z such that

(1) O<q~< 1, spt q~ N (/~U B) = 0 , Qo L (p#O.

Let

(2) ~ t = ( 1 - t ~) ~ ( O < t < 1)

AREA M I N I M I Z I N G CURRENTS BOUNDED, ETC.

We claim that

(3) there is a t such that i! S E ~I.A (Z), S is Orminimizing, and

spt (O S - O Oo)cB,

Otherwise there would be sequences

h , t2, t3 ,...--" 0,

then spt S N spt q~ = 0 .

S, , $2 , & .... ~: ~,,A (Z),

such that S i in O,;-minimizing, spt(OSi--aOo)cB, and

(4) spt Si N spt 9 # 0 .

Let RE~,,A(Z) be (}-minimizing with spt(OR--aOo)cB. Then

((},Sj>-~ ((},R)

because

41

((}, S i )< ( 1 - 0 -I <O, i , Sj) < ( 1 - 0 -1 ((},,, R ) <

_<(1--ti) -I ((},R)

by (2) and the minimizing property of St. Now the argument of [1, p. 522] produces an SE~I.A(Z), such that S is O-minimizing, spt (aS--aOo)cB, and for some ro satisfying 0<r0<dis t (B, sptq~), for some subsequence of i---- oo,

~:A [ (S- -S , ) L_ H (ro)] --* 0.

Here H(ro)={zEZ: dist(z,B)>ro}Dspt~. We note that

spt (S L H (ro)) N spt q~ = spt S N spt tp c E N spt q~ = 0 .

A lemma of Hardt [4,3.5] now implies that

sptSi N spt q~=spt (Si L H (rod N spt q~= O

for all but finitely many integers i. This contradiction of (4) establishes the claim (3).

Now let t, S be as in (3). We show that

r (S) = r (S)-< r (Oo) <(} (00).

42 FRANK MORGAN

The first equality and the last inequality follow from the definition (2) of Or,

(3), and (1). The first inequality follows from Lemma 2.1. From this contra-

diction of the minimizing property of Oo, we conclude that sptOoc/~.

3. The example.

Here we give quite explicitly the example described and discussed in the

introduction.

THEOREM 3.1. Given any positive integer n, there exists an integral current

T E 12 (IR ~) such that

(1) spt T-sp t 0 T is a smooth, two-dimensional mani/old, o / n components,

c2) o(llrll,x)--1 /or all x E s p t T - - s p t O T ,

(3) C=(1/n)OTEII([R4),

(4) spt C is a smooth curve o/ n+ 1 components,

(5) | ~or all xEsptC, and

(6) (l/n) T is the unique mass-minimizing normal current with boundary C.

Proo/. Fix a positive integer n. Identify I R ~ C 2. Define

g: C2• [0, 1] ---- ~2,

g (z, w, s) = gs (z, w) = (e 2~s z, e :~i'~ w) ,

r={g~: s E [0, 1) } = U C2),

Z = { ( z , w ) ~ C2: w > O } ,

A={(z, w)EZ: 1/3_<w--<3},

B = { ( z , w ) E Z : 1/3<_w<---1/2 or 2 < w < 3 } .

Let ~E6'**(IR) such that 0<~_<1, sp t~c (1 /3 ,3 ) , sp t (1 - -~ )N[1 /2 ,2 ]= = ~ . Define ~bEC**(C 2) by

(z, w)=~ ([w[).

AREA MINIMIZING CURRENTS BOUNDED, ETC. 43

Define a positive elliptic integrand ~ of degree one on Z by

a, (z, w, =) = (2 ~/n) V (n ~ l w l ~ + IzI ~) I~I ~ - (= . i z) ~ .

Noting that g, Z N g t Z = f ~ when s,t are distinct elements of [0, l /n) , one

checks (cf. [1, p. 363]) that if XfiNI(Z),

(7) M (g, (X• [0, l /n ] ) )=d~ (X).

r gives a metric on Z. Choose a > 0 such that p=(a, 1) has a normal spherical

neighborhood in Z which contains the points

qj= (0, 1 +]a) O<_i<n.

w

q2

ql G2GI

qo p

i - I

I I z

F igure 3.1 ( 8 )

Cose n = 5

See [5, p. 33] and our figure 3.1 (8). Let G i be the unique oriented geodesic from qi to p. If 0 _ < r n < n - 1 and u=e ~i"/", then

(9) g1~* Gt is the unique geodesic from qi to g=p.

4 4 FRANK MORGAN

Let

G=~,{gu, Gi: u=e2~i'~/"; i,m~.{O ..... n--l}},

See figure 1.1(1). We claim that

(lO) among all G's with spt(OG'--OG)cB, G minimizes ( ~, G). Furthermore, any other such

O-minim~,zing G" satisfies spt G 'cspt G.

To prove the claim, we recall that, for any such O-minimizing G', spt G ' -

spt0 G'--B consists of smooth, disjoint geodesics [1, 5.3.20]. By choice of a and the minimizing property of G', there are n geodesics, counting multipli-

cities, terminating at each g, p and originating at the q/s (not originating in B).

These n 2 geodesics comprise G', and by (9) sptG'csptG. Now n geodesics originate at each qi, all n are of the form g. Gi, and (O, guGi)=(O, Gi); therefore,

II (O,G')-- ~ n(O, Gi)---(~,G).

j=l

Hence G minimizes (O, G). Consequently

(11) the n 2 geodesics comprising G intersect only at the bounding points.

Now put

T t=g . (GiX [0, 1]),

(12) T= s T/=g. (GX[O,I/n]).

One easily verifies (1)-(5), using (11). In particular,

(13) 0 Tj=g. ((~p--~q)X [0 , 1 ] ) ,

(14) C=(1/n) 2 0 Tt=g, (g,X [0, 1] -- ~ gqi X [0, l /n]). ,/=1

To prove (6), let PEN(IR *) be mass-minimizing with OP=C. Let

AREA MINIMIZING CURRENTS BOUNDED, ETC. 45

1 O= f gs* P ds.

,=0

O is I" invariant, and a O=C. Since from its definition M (Q)_<M (P), O is mass-minimizing and M ( O ) = M ( P ) . Since go* P=P" it follows that sp tOD

~sp tP . For s{~ [0, l / n ] , let

O,=(O L_ ~b, argw, e2~i'+ ).

Since a priori O, EN~(IR 4) for almost all s [1, 4.2.1] and O is invariant under F, therefore O '=gs* Q~ for all sE[O, 1/n]. Also

(15) O L ~ = g , (Oox [o, l /n ] ) .

We now verify this sequence of equalities and inequalities:

(16) (G) = M (T) >-- M (n O) >..>_ M (n O L ~) = ~ (n O0).

The two equalities follow from (7), (12), and (15). The first inequality holds because n O is mass-minimizing with a(nO)=nC=OT. The second holds because 0_<t~_< 1.

Next we verify the opposite inequality ~(G)<O(nOo). We note that

OG=n(C, argw, 1 + )

by (9) and (14) and hence that

spt (0 G-- a n (2o) cB .

The inequality now follows from (10) and Lemma 2.1. Consequently, all the inequalities in (16) are equalities, Q0 is ~-minimizing, and O L, dT=O. Applying (10) and Lemma 2.2, we deduce that sp tOocsp tG. Therefore, by (12) and (15), s p t O ~ s p t T . We conclude that s p t P c s p t T . Since aP=C, it follows first, from (1), (2), and [1, 4.1.31], that P=Z~,iTi for some real numbers kj; and second, from (13) and (14), that Li=l/n and P=(1/n)T.

REFERENCES

[1] Federer H., Geometric measure theory, Springer-Verlag, Heidelberg & New York, 1969. York, 1969.

[2] Federer H., Real flat chains, cochains and variational problems, Ind. U. Math. l., :14 (1974), 351-407.

4 ~ FRANK MORGAN

[3] Gulliver R., Morgan F., The symmetry group o] a curve preserves a plane, Proc. A.M.S., 84 (1982), 408-411.

[4] Hardt R. M., On boundary regularity ]or integral currents or flat chains modulo two minimizing the integral o] an elliptic integrand, Comm. in P.D.E., 2 (1977), 1163-1232.

[5] Helgason S., Di]]erential geometry, Lie groups, and symmetric spaces, Academic Press, New York, San Francisco, London, 1978.

[6] Morgan F., A smooth curve in ]R 4 bounding a continuum o] area minimizing sur]aces, Duke Math. ]., 43 (1976), 867-870.

[7] Morgan ]. W., Sullivan D. P., The transversality characteristic class and linking cycles in surgery theory, Annals of Math., (2), 9 (1974), 463-544.

[8] Young L.C., Some extremal questions [or simplicial complexes V. The relative area o[ a Klein bottle, Rend. Circ. Matem. Palermo, (II), 12 (1963), 257-274.

Pervenuto il 29 luglio 1982

Massaehussetts Inst. o] Techn. Department o] Math. 2-181

Cambridge, MA 02139 U.S.A.