A Smooth Curve in R 3 Bounding a Continuum of Minimal Manifolds

FRANK MORGAN

Communicated b'y G. STRANG

Introduction

In 1950, after giving a heuristic argument for the existence in R a of infinitely many minimal surfaces with a common boundary, COURANT ([-4, p. 122]; see also NITSCHE [10]) blamed such "paradoxical phenomena" on permitting "such abstractions as general rectifiable curves" as boundaries. As to the possibility of a continuum of minimal surfaces with a common boundary, COURANT em- phasized that there were not even examples "to indicate plausible answers." In 1956 FLEMING [7] substantiated COURANT's heuristic argument with an exam- ple of a rectifiable (but not smooth) curve bounding infinitely many minimal surfaces (described and pictured nicely in [2, pp. 3-4]). Finally, in 1978 HARDT and SIMON, using work of TOMI's, further substantiated COURANT'S argument with the following result [8, Theorem 12.2]"

(1) A C4'~(e>0) Jordan curve in R 3 bounds only finitely many oriented area minimizing surfaces.

We now give a first example of an analytic curve (of four components) in R 3 which bounds continua of oriented minimal manifolds of arbitrarily large genus.

Fig. 1. One of a continuum of minimal surfaces.

Archive for Rational Mechanics and Analysis, Volume 75, �9 by Springer-Verlag 1981

194 F. MORGAN

We want to thank Professor ED MILLER for the extension to examples of arbitrarily large genus. It remains an open question whether a smooth Jordan curve in R 3 can bound infinitely many minimal surfaces.

Finally, we mention two earlier examples in R 4. In [-9] we gave an example of a smooth Jordan curve in R 4 which bounds a continuum of unoriented area minimizing surfaces. Secondly, E. BOMBIERI has told me that in [3, Theorem 2] he treats an example of countably many oriented minimal hypersurfaces in R 4

with a common smooth boundary.

The continua of minimal manifolds

Let R > 20, and let B be the curve in R 3 of four circular components given in cylindrical coordinates r, 0, z by

r=R,z=+_l; r = R _1, z=0 .

z I I

�9 -!_1 �9 I

. . . . - ~ l - e ~ o - ( - - ~ - - - T �9 -4--1 �9

- R I R I

Fig. 2. A cross-section of B in the x - z plane.

B is invariant under many isometrics of R 3, e.g. the group F of rotations about the z-axis and the rotation j through 180 ~ about the x-axis.

Theorem

For every natural number N, there is a continuum of oriented, minimal, embedded manifolds with topological boundary B and Euler characteristic at most - 2 N .

Proof. Consider the curve B' of two components

B'x: r=R, z = l ;

B 2 : r = R + I , z=0 , y > 0 ,

r = R - 1 , z=0 , y < 0 ,

y = z = 0 , R - I < [ x I < R + I

B~

Fig. 3. B' viewed from above.

Continua of Minimal Manifolds 195

Define the N to 1 covering map

p : R 3 - - {z-axis} ~ R 3 - {z-axis}

p(r, O, z)=(r, NO, z).

An integral current S can be lifted uniquely to an integral current S, and p . =NS. Let F be the elliptic integrand on R 3 - {z-axis} such that

F (S) = M (S).

Let S be an F minimizing integral current with respect to {(r, O, z)~R3: 1 < r < 2 R } with boundary B', where B' is oriented so that F(S) is as small as possible. Finally put

~ ' = spt S wj(spt S ) -B .

Lemma. (1) M(S)=<4rtR;

(2) spt S c {(r, 0, z)eR ~ : R/2 < r < 3 R/2} ;

(3) spt S is an embedded minimal manifold with boundary p - 1 B', except at the 2 N corner points

( R + I , k~/N,O) ( 0 < k < 2 N ) ;

(4) spt S is connected;

(5) spt ~ _ p - 1 B ' = {(r, 0, z)~R3: z>0}.

Fig. 4. p- ~ B' viewed from above, N = 6.

Proof. Consider the pieces (cf. Figure 3)

R <_r<_R + I,

R-I<_r<_R,

r~R~

surface T with boundary B', consisting of the three

0_<0<n, z=0 ;

~ < 0 < 2 ~ , z = 0 ;

0 < 0 < 2 ~ , 0_<z_< 1.

Here M ( S ) = F ( S ) < F ( T ) = 4 ~zR. But now if (2) fails, spt S intersects the cylinder r = R/2 or r = 3 R/2, and by monotonicity of the mass ratio for minimal surfaces ([1, 5.1(2), p. 445]), we have

F(S)=M(S)>=rt(R/2- 1) 2 >4r tR

196 F. MORGAN

because R>20. This contradiction of (1) proves (2). Since S is locally mass minimizing, the interior regularity follows from [6, 5.4.15, p. 644] and the boundary regularity from [8, w or 11.1].

If S were not connected, each component, as a minimal surface with a planar boundary, would be planar, and we would have M(S)=2zcR2+~, contradicting (1). Now (5) follows by the maximum principle.

Completion of proof of the theorem. By (5), spt S - B and its 180 ~ rotation about the x-axis j(spt S ) - B intersect in the 2 N segments of p-1 B ' - B given by (cf. Figure 4)

(6) R - I < _ r < R + I , O=k~/N, z = 0 ( 0 < k < 2 N ) .

Along the segments 0 = 0, ~z/N (and along the 2 N - 2 others by symmetry), they fit together to form a minimal manifold

Jg = spt S wj(spt S ) - B

by the reflection principle [5]. One notes that ~ ' is orientable and has B as boundary both topologically and as an integral current. Furthermore, from (4) it follows that , # intersects the plane z =0 in the 2 N segments of p - I B ' - B (6). Therefore if a~, a z are distinct rotations in F of less than rc/2N radians, then 0.1 ~/t + 0.2 ~', and we have a continuum of minimal manifolds with boundary B.

The Euler characteristic Z of a connected, oriented, two-dimensional man- ifold Jr or without boundary is defined via homology:

X(~') = rank H 0 (dr rank H 1 (Jg) + rank H 2 (J/()EZ U { - - O0 }.

We will need the fact t ha t / f JV is an N-fold cover of dg, then

(7) Z ( X ) = N Z(~'),

even when J/g is a noncompact manifold. Let D be a triangulation of ~g, a n d / ) its lift to JV. As is well known, if ~ is compact (and hence D is finite) then

)~(J//) =no. of vert ices- no. of edges+ no. of faces,

and (7) follows easily. It ~ is a noncompact manifold, one first checks that if E is a finite subcomplex of D, there is a finite subcomplex E', E c E ' c D such that the natural homomorphism HI(E')--*HI(D) is injective. Hence there is an in- creasing exhaustive sequence of finite subcomplexes

Dl cDz cD3 ~. . . ~D

such that H1 (Di) ~ H~ (D) is injective; hence rank H 1 (Jr = lim rank H~ (Di). One i ~ o ~

can further require that each D~ and its lift /)i to JV are connected, so that rankH0(Di)=rankHo(Di)=l and rankHz(Di)=rankH2(/)i)=O. Finally one checks that H 1(/)i) ~ H l ( b ) is injective. Hence,

Z(~') = lira Z(b,) = lira N z(Di) = N Z(~Cg).

Continua of Minimal Manifolds 197

N o w to estimate the Euler characteristic of the manifold ~ ' , we note that the manifold

A/" = spt S wj(spt S) - B

has Euler characteristic at most - 2 because its topological boundary has four components and is connected (by (4)). Since JC{ is just the lifting of JV', we thus get

X (//{) = N ;((~A/') < - 2 N.

This work has been partially supported by NSF Grant MCS-7621044.

References

1. ALLARD, W.K., On the first variation of a varifold. Annals of Math. 95, 417-491 (1972).

2. ALMGREN, F. J., JR. Plateau's Problem. New York: W. A. Benjamin 1966. 3. BOMBIERI, E., Recent progress in the theory of minimal surfaces, l'Enseignement

Mathematique 25, 1-8 (1979). 4. COURANT, R., Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. New

York: Interscience 1950. 5. DOUGLAS, J., The analytic prolongation of a minimal surface over a rectilinear

segment of its boundary. Duke Math. J. 5, 21-29 (1939). 6. FEDERER, H., Geometric Measure Theory. New York: Springer 1969. 7. FLEMING,W.H., An example in the problem of least area. Proc. AMS 7, 1063-1074

(1956). 8. HARDT, R., & SIMON, L., Boundary regularity and embedded solutions for the orient-

ed Plateau Problem. Ann. of Math. Studies 110, 439-486 (1979). 9. MORGAN, F., A smooth curve in R 4 bounding a continuum of area minimizing

surfaces. Duke Math. J. 43, 867-870 (1976). 10. NITSCHE, J. C. C., Concerning the isolated character of solutions of Plateau's Problem.

Math. Z. 109, 393-411 (1969).

Department of Mathematics Massachusetts Institute of Technology

Cambridge

(Received May 10, 1979)