NON-EXISTENCE, NON-UNIQUENESS AND IRREGULARITY OF SOLUTIONS TO THE MINIMAL SURFACE SYSTEM
BY
H. B. L A W S O N , J r . a n d R. 0 S S E R M A N ( 1 ) University of California, Stanford University, Berkeley, California, USA Stanford, California, USA
Table of contents
w 1. I n t r o d u c t i o n . . . 1
w 2. The minimal surface s y s t e m . . . 2
w 3. A brief s u m m a r y of k n o w n results in eodimension one . . . 4
w 4. The existence of solutions of dimension two . . . 5
w 5. The non-uniqueness and n o n - s t a b i l i t y of solutions of dimension two . . . 6
w 6. The non-existence of solutions in dimensions >/4 . . . 11
w 7. The existence of non-parametric m i n i m a l cones . . . 13 w 1. Introduction
T h e s t u d y of r e a l - v a l u e d f u n c t i o n s whose g r a p h s are m i n i m a l surfaces h a s a long a n d rich h i s t o r y , a n d b y n o w we h a v e a f a i r l y p r o f o u n d u n d e r s t a n d i n g of t h e s u b j e c t . I n c o n t r a s t , a l m o s t n o t h i n g is k n o w n a b o u t v e c t o r - v a l u e d f u n c t i o n s whose g r a p h s a r e m i n i m a l surfaces, a n d t h i s p a p e r s h o u l d e x p l a i n , a t l e a s t in p a r t , w h y t h i s is so. I t will be s h o w n t h a t m a n y of t h e d e e p a n d b e a u t i f u l r e s u l t s for n o n - p a r a m e t r i c m i n i m a l surfaces in codi- m e n s i o n one fail u t t e r l y in h i g h e r c o d i m e n s i o n s .
W e shall be c o n c e r n e d p r i m a r i l y w i t h t h e D i r i c h l e t p r o b l e m for t h e m i n i m a l s u r f a c e s y s t e m on a b o u n d e d , c o n v e x d o m a i n ~ in R n. U s i n g a n o l d a r g u m e n t of R a d o , we s h a l l s h o w t h a t for n = 2 , t h e D i r i c h l e t p r o b l e m is s o l v a b l e for a r b i t r a r y c o n t i n u o u s b o u n d a r y d a t a . H o w e v e r , we t h e n c o n s t r u c t e x a m p l e s t o s h o w t h a t t h e s e s o l u t i o n s are not unique i n general. Moreover, we shall s h o w t h a t s u c h surfaces need not even be stable i n c o n t r a s t w i t h t h e f a c t t h a t i n c o d i m e n s i o n one, n o n - p a r a m e t r i c m i n i m a l surfaces a r e a b s o l u t e l y a r e a m i n i m i z i n g .
W e shall t h e n show t h a t for n >~ 4, t h e D i r i c h l e t p r o b l e m is not even solvable i n g e n e r a l .
(1) Work partially supported by NSF grants MPS-74.23180 and MPS 75.04763.
1 - 772904 Acta mathematica 139. Imprim6 le 14 Octobr9 1977.
2 H, B. LAWSON~ JR. AND R, OSSERMAN
I n fact, for each n ~ 4 it is possible to find a C ~ f u n c t i o n / : S ~ - l ~ I t k for some integer k, 2 ~</c ~< n - 2, with the p r o p e r t y t h a t there are no Lipschitz solutions F to the minimal sur/ace system in D n such t h a t F i s h - 1 = / . Moreover, t h e same s t a t e m e n t holds for a l l / ' in a large C 1 neighborhood of /. Such a family of examples can be g e n e r a t e d for each non-trivial element of ~_~(S~-~).
Finally, b y e x a m i n i n g certain specific cases in detail we are able to show the existence of n o n - p a r a m e t r i c minimal cones. I n particular, this gives examples of Lipschitz solutions to the minimal sur/ace system which are not C ~, t h e r e b y m a k i n g sharp the basic r e g u l a r i t y result of Morrey which states t h a t a n y C 1 solution is real analytic.
We would like to t h a n k Bill Allard for several i n f o r m a t i v e conversations relating to this work.
w 2. The minimal surface system
Let ~ be an open set in R n a n d consider a C 2 immersion F : ~ R n+k. T h e n F is a minimal immersion if a n d o n l y if F satisfies the s y s t e m
Y =o, (2.1)
~,~laX ~x
where g = d e t ((g~j)), ((gtj))=((gtj))-i a n d g~j=(OF/Ox ~, ~F/~xJ). This is equivalent to t h e requirement t h a t F ( ~ ) h a v e m e a n c u r v a t u r e identically zero.
The immersion F is said to be non-parametric if it has t h e form F(x)= (x,/(x)) for some function [: ~ R k. I n this case the system (2.1) has the form:
~ l ~ x ]= . . . , n
".-, g) [ ] / - , j c~/~ (2.2)
where g a n d ((g~J)) are defined as above, a n d in this case
\
This is clearly equivalent to t h e system:
~-IOX ] ~ " ' ' g n
(22)
" . t j O ~ I ^
[ ~ ~ - ~ = u . t f , , ~ - I (].~ u . o
T H E M I : N I M A L S U R F A C E S Y S T E M
F o r functions of class C 2, this system can be replaced b y the smaller one:
n a
To see this we note t h a t (2.3) simply expresses the fact t h a t the vertical projection of the mean curvature vector to the graph of / is identically zero.
B y a somewhat more subtle argument, it is shown in [30] t h a t the system (2.2) (or (2.2)') for C 2 functions is also equivalent to:
n 2
5 g'J ~ / - 0 (2.4)
Note t h a t the system (2.2) (or (2.2)') is defined in the weak sense for a n y locally Lip- schitz function / on ~ . I n this case (2.2) is equivalent to the condition t h a t the first vari- ation of area of the graph of / with respect to smooth, compactly supported deformations of ~ • R k, is zero. I n other words, the graph of / is a stationary integral varifold in the sense of Almgren (of. [3]). This leads naturally to the following.
CONJECTURE 2.1. The systems (2.2) and (2.3) are equivalent for a n y locally Lipschitz function / in ~ .
Suppose now t h a t ~ is bounded and strictly convex and t h a t d ~ is of class C r for r >~ 2. For the remainder of the paper we shall be concerned with the following.
D I R I C H L E T P R 0 B L E M. Given a / u n c t i o n r d ~ ~ It k o/class C s, 0 -~ s <~ r , / i n d a / u n c t i o n / E C ~ N Lip (~) such that / satisfies the m i n i m a l sur/ace system (2.2)in ~ and ] [ ~ = r
When s >~ l, we/urther require that the area o / t h e graph o / / b e / i n i t e .
The study of the Dirichlet problem usually falls into two distinct parts, those of the existence and the regularity of solutions. I t is an unusual fact t h a t for the non-parametric minimal surface system more is known about regularity t h a n existence.
T~EOREM 2.2. (Interior regularity; C. B. Morrey [22], [23]). A n y C 1/unction / which satis/ies the system (2.2) is real analytic.
THEOREM 2.3. (Boundary regularity; W. Allard [2]). Suppose that r is o/ class C s'~ /or 2 ~ s ~ ~ or s = w, and let / be any solution to the Dirichlet p r o b l e m / o r r in ~ . T h e n there is a neighborhood U o / d ~ such t h a t / e C S ' ~ ( U fl ~ ) .
Note. Theorem 2.3 is deduced from the work in [2] as follows. Since ~ is strictly convex and r is class C ~, the arguments in 5.2 of [2] show t h a t the graph F I of / in R n x R k, at each point of its boundary, has a t a n g e n t cone consisting of a finite n u m b e r
H . B. LAWSON~ J R . AND R . 0 S S E R M A N
of half-planes (of dimension n), each of which projects non-singularly into R ~. Since each such cone is the limit of a sequence of dilations of FI, it is easy to see t h a t the cone consists, in fact, of a single half-plane. I t follows t h a t the density
O(llr111,
x)= 89 at each boundary point x. Theorem 2.3 is then an immediate corollary of the Regularity theorem in [2, w 4].Remark. Perhaps it should be pointed out t h a t if ~ is not convex, the Dirichlet problem is not necessarily solvable in general even for n = 2 and r = s = w . I n fact when n = 2, convexity is necessary and sufficient for a solution to exist corresponding to arbitrary C s boundary values [12]. When n > 2, convexity is still sufficient, but the precise necessary and sufficient condition is t h a t the mean curvature of the boundary with respect to the interior normal be everywhere non-negative [17].
w 3. A brief s n m m a r y of kllown results in c o d l m e n s i o n o n e
For comparison with our later results in higher codimension we list here some of the facts known about non-parametric minimal surfaces in codimension one.
T H E o R E M 3.1. For k = 1, the Dirichlet problem is solvable /or arbitrary continuous boundary data. Furthermore:
(a) The solution is unique.
(b) The solution is real analytic.
(c) The solution is absolutely area minimizing, i.e., its graph is the unique integral current o/least mass in R "§ /or the given boundary (the graph o/4).
The existence for C ~ boundary data follows from Jenkins and Serrin [17], and for general continuous boundary data from the a priori estimates in [5]. The regularity of Lipschitz solutions follows from de Giorgi [7]. (See [37] or [25].) The remainder of the theorem is classic. (See [21, pp. 156-7] for part (c).)
For the purpose of completeness we mention the following strong removable singula- rities result.
THEOREM 3.2. Let K be a compact subset o] ~ with Hausdor]/ ( n - 1 ) . m e a s u r e zero.
Then any (smooth) /unction /: ( ~ - K ) - ~ R which satis/ies the minimal sur/ace equation (2.2)' in ~ - K extends to a (smooth) solution in all o / ~ .
This was first proved b y Bers [4] for n = 2 and K = {point}. The general 2-dimensional case was proved by Nitsche [27]. The result in arbitrary dimensions is the work of de Giorgi and Stampaeohia [8].
Note. I t has been known for some time t h a t this last result is not true in general
T H E M I N I M A L S U R F A C E S Y S T E M
codimensions. In fact, in [28] an example is given of a bounded function /:{(x, y)ER:
0 < x 2 + y~ ~< 1 }-* R 2, whose graph is a minimal surface in R 4, b u t which does not extend contin- uously across 0. However, under certain (necessary) restrictive hypotheses, removable singularity results can be proved in higher codimension (el. [14], [31]).
w 4. The existence of solutions of dimension two
The following result is not new. I t is an immediate consequence of old techniques of T. Rado and has certainly been known to the authors for some time. (See, for example [29, Thm. 7.2].) We include a proof here because the arguments involved will be useful in the next chapter.
THEOREM 4.1. For n = 2 and any k>~l, there exist solutions /EC~(~)N C~ to the Dirichlet problem/or any given continuous boundary/unction r d ~ R k.
Proo/. We begin by recalling the notion of a generalized parametric minimal sur/ace in R" having a given J o r d a n curve 7 as boundary. This is a map ~o: A ~ R ", where A = {z = x + i y e C: [z I ~< 1 }, with the following properties:
(a) ~ e C(A) n C~(AO),
(b) I ~ x l ~ = l ~ l ~ and <yJx, v/y>=0,
(c) A~=0,
(d) ~[ da: dA-~? is a homeomorphism.
The fundamental work of Douglas and Rado asserts the existence of such a minimal surface for any given J o r d a n curve 7 = RN (of. [9], [32] or [6]).
Theorem 4.1 will be an immediate consequence of the following result.
THEOREM 4.2. Let ~ be a Jordan curve in R 2+~ and suppose that ~, can be expressed as the graph ol a continuous/unction r d ~ - ~ R k where ~ is a bounded, convex domain in R 2.
Then every generalized parametric minimal sur/ace with boundary ~ has a one-to-one, non- singular projection onto ~ , i.e., every such sur/ace can be expressed as the graph o / a / u n c t i o n /: ~ R k where/eC~ N C~(~) and / satis/ies (2.2) in ~ .
To prove this we shall need the following basic result of T. Rado [33]. (Rado states the theorem for N - - 3 , however the proof works in general, cf. [19], [29].)
PROPOSITION 4.3. Suppose v2: A ~ R N is a generalized parametric minimal sur/ace with boundary 7, where 7 is an arbitrary Jordan curve in R ~. Let ~: RN-~R be a linear/unction
6 H . B , L A W S O ] N , J R . A N D R . OSS:ERMA~[
and suppose that at a point (x0, Yo)EA0, the function ~o~ has a zero o/order k, i.e., 8x,~yy..~(~oyJ ) = 0 for i+]<~k.
(x0, Y0)
Then the hyperplane H = { X E R~: ~t(X) = 0} meets V in at least (2k + 2) connected components.
Roughly, this proposition follows from the m a x i m u m principle and the observation t h a t the zeros of ~toy~ in a neighborhood of a zero of order k have the structure of the zeros of a harmonic homogeneous polynomial of degree k + 1.
Proof o/ Theorem 4.2. L e t ~: R 2 • R k ~ R 2 be projection and consider the m a p ~o~v:
A - + ~ . (Note t h a t zro~v(A~ b y the strict m a x i m u m principle.) Suppose t h a t at some point (x0, Y0)EAO we had rank(d(~ro~v))~<1. Then for a non-trivial linear function of the form ),(x, y, zl, ..., z~) =2(x, y ) = a x + by, we would have that/~oyJ has a critical point of order >~ 1 at (x0, Y0). I t then follows from Proposition 4.3 t h a t the hyperplane H defined b y ~t = ~(x0, Y0) m u s t intersect ~ in at least 4 components, ttowever, b y the convexity of
~ , H N? has exactly two components. I t follows then t h a t ~oy~: A ~ is a local diffeo- morphism which extends to a homeomorphism of the boundaries. I t follows easily t h a t
~zo~p is one-to-one. This completes the proof.
We now state another immediate consequence of the arguments above. Given a linear function ~t on R N and c e R, let Hc(~t ) = {X e RN:
~(X)=c}.
PROFOSITION 4.4. Let ~ be a Jordan curve in R N and suppose ),: RN-+R is a linear function such that Hc(~ ) N ~ has at most/ice components for each c E R. Then for any generalized parametric minimal surface yJ: A ~ R tr with 7, as boundary, the function ,~ov 2 has only non-
degenerate critical points in A ~ and those critical points have index 1.
w 5. The non-uniqueness and non-stability of solutions o| dimension two I n this section we shall prove the following result. L e t
D = {(x, y)ER2: x ~ +y2 ~< 1}.
THEOREM 5.1. There exists a real analytic function r dD--~R 2 with the property that there exist at least three distinct solutions (each of class C '~) to the Dirichlet problem/or r in D.
Moreover, one of these solutions represents an unstable minimal surface; that is, the area of the graph can be decreased by arbitrarily small deformations which/ix the boundary.
Proof. L e t ~ denote the graph of r in R i. I t will suffice to show t h a t there exist two geometrically distinct parametric minimal surfaces with b o u n d a r y 7', each of which has
THE MINIMAL SURFACE SYSTEM
the following additional property:
The surface minimizes the area and Dirichlet integrals among all piecewise differentiable maps of A into Euclidean space which carry dA homeomorphically
onto 7. (5.1)
If we succeed in proving this, then b y the theorems of Morse-Tompkins [24] and Shiffman [35], there is a third, unstable parametric minimal surface with 7 as boundary. B y Theorem 4.2 every generalized parametric minimal surface with b o u n d a r y 7 is, in fact, a non- parametric surface over D and we will have proved the theorem.
For clarity of exposition we shall begin with an example in codimension 3. We shall then indicate how similar arguments will produce examples in codimension 2.
Consider the regular, C ~~ J o r d a n curve 70 in R a pictured in Figure 1. This curve lies in the union of two pairs of parallel planes, each pair a distance e a p a r t where s < l . The curve is assumed to be invariant under the s y m m e t r y a0(x, y, z ) = ( - x , - z , y). Note t h a t
~olw preserves the orientation of 70.
z
Y
F i g u r e 1
Consider now the m a p p i n g r dD~7o given b y (the appropriate multiple of) arc- length, and set r = Re0 for some R ~ 1. We t h e n define 7 to be the graph of ~ in R 5. Note t h a t since ao is an isometry and r is an arc-length parametrization, we have
ao(r v)) = r - v , u) Hence 7 is invariant under the s y m m e t r y
a(u, v, x, y, z) = ( - v , u, - x , - z , y ) .
H. B. LAWSON~ J R . AND R. OSSERMAN
U p to a scalar change, ~ represents an ever-so.slight deformation of ~0 into five-space.
Our first m a j o r claim concerning the curve 7; is the following.
PROPOSITION 5.2. Let v2: A-~R 5 be any generalized parametric minimal sur/ace with boundary ~, and/et ~: RS-~R be the coordinate/unction ,~ =x. Then ~ov 2 has exactly one critical point in A and that critical point is non-degenerate.
Proo/. Each hyperplane Hc(~)ffi{XER6:2(X)=c} clearly meets ~ in at most four points. Hence, b y Proposition 4.4 every critical point of ~toy~ in A0 is non-degenerate and of index 1.
We now consider the corresponding non-parametric s u r f a c e / : D - ~ R a guaranteed b y Theorem 4.2. I f re: RS-~R ~ denotes projection onto the first two coordinates, then h = reoy): A - ~ D is a homeomorphism a n d we h a v e t h a t
~(z) = (h(z), /(h(z)))
for all zEA. Since h is a diffeomorphism on the interior of A and since ~o/oh =2oyJ, we see t h a t the function ~to/has the p r o p e r t y t h a t each of its critical points in D O is non-degenerate and of index 1.
When restricted to the boundary, the function )~o/Id~ =~tor has exactly four critical points, two non-degenerate m a x i m a a n d two non-degenerate minima.
We are now in a position to a p p l y elementary Morse theory to ~to/. F o r each c E R we consider the sublevel set Dc={(x, y)ED: 2o/(x, y)<~c}. For c slightly larger t h a n c o
= i n f D (~to/}, Dc consists of two components, each homeomorphic to a disk. As c increases, we add a one-handle to this manifold for each critical point in D ~ Since D ~ = D, we con- clude t h a t there m u s t be exactly one critical point of ~to/in D ~ I t follows t h a t ~ o ~ has exactly one critical point in A0. This completes the proof of the proposition.
Figure 2
F r o m the work of Douglas [9] we know t h a t there exists a t least one parametric minimal surface ~: A - ~ R 5 having b o u n d a r y ~ and satisfying condition (5.1). L e t us suppose t h a t there are no other such surfaces which are geometrically distinct from yJ (i.e., which
THE MINIMAL SURFACE SYSTEM
Y
a-invariant surface
Y
Comparison surface
]Figure 3
are not just a reparameterization of the same surface). Then since ~: RS-~R 5 is an isometry which preserves 7, we see t h a t a o ~ is again a parametric minimal surface having boundary and satisfying condition (5.1). Hence, ~ and a o ~ must have the same image surface, say ]E, in RS; t h a t is, a ( 2 ] ) = ~ . B y Proposition 5.2 the linear function ,~=x has exactly one critical point ~o when restricted to ~.o. I t follows that a(p) = p , and so p =0.
We will now show that the surface r / c a n n o t satisfy condition 5.1 for R sufficiently large. In fact we claim t h a t the area of ~ satisfies the inequality
A(~,) >~ 4~R 2 (5.2)
whereas one can easily construct a surface ~,' with boundary 7 such t h a t
A (Z') ~< [2~ + 2 (~ + l) e] R 2 + 0 (R) + ~ (5.3) where I is the length of the four parallel arcs of 7 (ef. Fig. 1). To construct this surface one first chooses a parametrization ~o: D-~RS of the comparison surface pictured in Figure 3, with ~01dD=r The area of the comparison surface is bounded above by 2 z + 2 ( ~ + / ) . The surface ~.' is then defined to be the graph of the map yJ = RVJ 0. I t easy to verify the inequality (5.3).
To deduce (5.2) we consider the hyperplane
H={XeRS: ,~(X)=0}
and observe that Y . - H decomposes into four connected components, (each parameterized by ~ restricted to one of the four wedge-shaped regions in the middle picture of Figure 2). Denote these components by Z1 ... ~'4, and observe t h a t the boundary of each :E~ maps b y orthogonal projection onto the boundary of a key-shaped region R~ in either the (x, y)-plane or the (x, z)-plane. Consequently, each Z~ maps b y orthogonal projection onto R~. Since A(R~) > ~ R 2, it follows t h a t A(Z~) ~>gR ~ for each i and (5.2) is established.We have shown t h a t there must be two geometrically distinct parametric minimal surfaces with boundary 7, which satisfy condition (5.1). This establishes an example of the type claimed in the theorem with the exception t h a t the curve is class C ~176 and the codimen- sion is three.
10
V
H . B. LAWSO~N, J R . A N D R . 0 S S E R M A N
4o
J U
Y
F i g u r e 4
~ X
We now observe t h a t throughout the above a r g u m e n t it is possible to replace the curve y with a real analytic approximation. This is done b y replacing r with a sufficiently large piece of its Fourier e~pansion. Since the Fourier polynomials preserve such properties as
and
for (complex-valued) functions F on the circle, we have t h a t the analytic approximation will also be o-invariant.
We shall now indicate how to construct an example of codimension 2. The arguments are all essentially the same except t h a t one begins with a m a p r 1 6 2 where R>~I and 40: dD-~It~ is a suitably parametrized double tracing of the curve pictured in Figure 4, where each loop is traversed once in each direction, and where the p a r a m e t e r is a multiple of arc length.
The graph of r is invariant b y the symmetries ok: IO-~R 4, k = 1, 2 where 01(u, v, x, y) = ( - u , - v , x, - y )
a,,(u, v, x, y) -- ( - u , v, - x , y).
L e t ~ be the graph of r I f we again assume t h a t there are not two geometrically distinct minimal surfaces with b o u n d a r y ~ satisfying condition (5.1), then the surface ]E given b y the Douglas solution to the Plateau problem for ~ m u s t be invariant under o 1 and 02.
The linear function ~t = x has exactly one critical point p on ~]. This follows from an argu- m e n t entirely similar to the proof of Proposition 5.2. Consequently, r a ~ ( p ) = p , and so p = 0 . The rest of the argument proceeds as above. The appropriate comparison
T H E M I N I M A L S U R F A C E S Y S T E M 11 surface is given b y a m a p ~F = R~Fo where ~F 0 is chosen to be linear on the lines u -~- v = c.
For I c ] ~< 1, ~F 0 will be constant on these lines. The image of ~F 0 will be the curve together with the region bounded b y the loop on the left (covered twice). This completes the proof of Theorem 5.1.
w 6. The non-existence of solutions in dimensions ~ 4
The main result of this section is the following. L e t D~=(x~R~: Ilxll <1} and set S~ = dDn+l.
THEOREM 6.1. Let r S n + k ~ S n c R n+l be any C 2 mapping which is not homotopic to zero as a map into S ~, and suppose k > 0 . Then there is an Re such that/or each R >~ Re there is no solution to the Dirichlet Problem/or the boundary/unction ~ = R . r
Remark 6.2. I t follows from the Implicit function theorem (cf. [26]) t h a t the Dirichlet problem is always solvable for sufficiently small b o u n d a r y data. Hence, given r as in Theorem 6.1, there is an re > 0 such t h a t for all r, I r[ ~< re, the Dirichlet problem is solvable for boundary data Cr = r. r
Proo/. The proof of this theorem rests on the following two results. I n the statements, the t e r m minimal variety means a n y integral current T with compact support such t h a t the first variation of mass is zero with respect to a n y smooth deformation supported a w a y from the boundary of T. I n our applications we will only need to consider currents which are given as the oriented graphs of Lipschitz functions.
T H E O R E ~ A. Let V be any p-dimensional minimal variety in R N which is regular at the boundary (i.e., in a neighborhood o / s u p p (dV), V is given by an oriented, C 1 submani/old with boundary). Then the mass o/ V is given by the ]ormula
.~,l f d
31(V)=:, ( v , x ) -1 (6.1)
v
where x is the position vector with respect to any euclidean cordinate system and where ~' is the unit exterior normal/ield to d V along d V.
Note. The mass of V is the "weighted volume!'. I f V is a Lipschitz submanifold, 3I(V) is just the Hausdorff p-measure of t h a t submanifold.
We shall only need this result in the special case t h a t d V lies in a sphere about the origin in R N. Under this assumption the proof is quite easy, and the reader is referred to [13] or [20] for details.
12 H . B . L A W S O 1 % J R . A N D R . O S S E R M A I ~
TH~.OREZ~ B. Let V be a p-dimensional minimal variety in It s. Suppose that xoER N lies on V and that B(xo, R)={xERN: < R } ao~ ~ot meet the boundary o/ V. Then
M(V) >~ crR~ (6.2)
where c~ i8 the volume o/ the unit ball in R ~.
This is a direct consequence of the well known fact t h a t r Iq B(xo, r))/%r p is monotone increasing and lim~_,o r (See [10], [13] or [20].)
Let us now suppose that for fixed R, there is a solution/R to the Dirichlet problem for Ca. Let Vn be the minimal variety given b y the graph of ]n in R N where N --- 2(n + 1) + k.
B y Theorem 2.3 VR is regular at the boundary. We clearly have t h a t dVR = {(x, Rq~(x)):
[[x[[ = 1 } is contained in the sphere of radius l/l + R 2 about the origin in R N. Hence, it follows from (6.1) t h a t
l/i + R2 M(dVR). (6.3)
M(Vn) ~< n + k + ~
We now observe that ]R represents a homotopy of CR to zero in R n+l. Since CR is not homotopic to zero as a map into S n and since R n+1- {0} has the homotopy type of S n, we conclude that /n(D n+k+l) cannot be contained in R n+l-{0}. Hence, there must be a point xoED "+k+l such that/R(Xo) =0, i.e., there is a point of the form (x 0, 0) on V~.
We now have t h a t
dist [(x o, 0), dVa] = min ]/ x - Xo//2 + [IR~b(x)ll ~ > R.
kvl-1
Hence, we m a y apply Theorem B to VR in the ball of radius R about (x0, 0) and conclude that
M(Vn) >~ c~+~ + , R ~+k+ ' . (6.4)
One can easily see t h a t there is a constant c such t h a t M(dVR)<~cR n for all R > 0 . Combining this with the inequalities (6.3) and (6.4) gives the estimates
V1 + R 2
n + k + l
c R n .
Since k > 0, the theorem follows immediately.
Example 6.3. The simplest example of a map satisfying the hypotheses of Theorem 6.1 is the Hopf map ~: S s ~ S ~ given b y
~(z~, z~) = (] z~ J~ - J z2 J~, 2z~ ~) (6.5)
THE MINIMAL SURFACE SYSTEM 13 where S s is considered as the unit sphere in C 2 = R 4 and S 2 as the unit sphere in R • C = R s.
I n this case we can m a k e a fairly explicit calculation of R~.
Observe t h a t the group SU(2) acts naturally on C 2 and on R 3 (as S0(3) =SU(2)/Z2).
The m a p ~ is equivariant with respect to these actions. This is most easily seen b y recalling t h a t S 2 =PI(C) and S0(3) = P U ( 2 ) , the group of isometries of the projective line. I t follows t h a t the graph of each ~ = R ~ 7, R > 0 , is invariant under the joint action of SU(2) on R 7.
Since SU(2) is transitive on S s, each such graph is an orbit of this action.
We now wish to compute the volume of the graph of ~R. B y the homogeneity it will suffice to compute the volume element at a single point. At a n y point x E S 3 we can choose an orthonormal basis el, e2, e 3 of Tx(S 3) such t h a t (~TR),(e3)=0 and (~R),(es), ~ = 1 , 2, are perpendicular and of length 2R. Hence the metric induced b y the graphing immersion x' ~-~ (x', ~R(x')) at x has m a t r i x
1 + 4R 2 0 )
1 + 4R ~
0 1
with respect to the basis el, e2, e a. I t follows t h a t the volume of the graph FR of ~R is vol (F~) = fs, (1 + 4R 2) *1 -- (1 + 4R ~) 2~ ~. (6.6) Therefore, combining the inequalities (6.3) and (6.4) in this case we get
~2 y~2
R 4 ~< M(/~) ~< ~ ~ R 2 (1 + 4R~).
I t follows t h a t there are no solutions to the Dirichlet problem/or ~R in D 4 whenever R >~ 4.2.
I t is interesting to note t h a t while there is no non-parametric minimal v a r i e t y with boundary FR for R>~4.2, there m u s t be some minimal v a r i e t y with this b o u n d a r y b y the basic work of Federer and Fleming [11]. I n fact there m u s t be such a v a r i e t y which is SU(2)-invariant [18]. This symmetric solution corresponds to a geodesic arc in the orbit space (cf. Hsiang and Lawson [16]). I t can be described topologically as (the closure of) the graph of the rational m a p Q: D4-~pI(C) given b y ~(Z1, X~)= [ZI/Z~]. I t is diffeomorphic to the oriented 2-disk bundle over S ~ of Chern class 1.
w 7. The existence o| non-parametric minimal cones
The existence of solutions to the Dirichlet problem for b o u n d a r y d a t a ~s = R . ~ when R is small, a n d the non-existence when R is large, lead one to suspect t h a t there should be a critical value R 0 for which there exists some sort of singular non-parametric
14 H . B . L A W S O N , J R . A N D R. O S S E R M A N
solution. I n the special case of t h e H o p f map, the s y m m e t r i e s indicate t h a t this singular solution should simply be the cone over the b o u n d a r y values. We shall show t h a t this is indeed the case.
Recall t h a t for a n y submanifold MY c S ~, the cone C ( M Y) = (tx E R~+I: x E M Y a n d t > 0) is a minimal v a r i e t y in R ~ 1 if a n d only if M Y is a minimal submanifold of S ~. F o r each zr 0 < ~ ~< 1 we consider the e m b e d d i n g i~: $3-~S 8 given b y
i~(x) = (~x, V1 - ~ ~(x)), (7.1)
where 9 : $ 3 - ~ $ 2 is the m a p given b y (6.5). E a c h submanifold i~(S 3) is an orbit of t h e action of SU(2) oil R: = C 2 • R 3 given b y the n a t u r a l diagonal m a p S U ( 2 ) ~ S U ( 2 ) x S0(3) (as we saw in the last section). I n fact, these orbits are principal orbits since SU(2) acts freely on them. N o w b y a basic result of W u - Y i H s i a n g [15] the principal orbits of m a x i m a l volume are minimal submanifolds of S 6. Therefore, we w a n t to c o m p u t e the v o l u m e function on t h e orbit space X =$6/SU(2).
We first observe t h a t t h e orbit space itself is highly symmetric. Recall t h a t SU(2) Sp(1)-~S a is just the g r o u p of u n i t quaternions, a n d the a b o v e representation on C 2 ~ H is just quaternion multiplication on the left. The group SU(2) also acts on C2~=H b y q u a t e r n i o n multiplication on the right. L e t ~ denote this representation, a n d e x t e n d it trivially to ~ =Q (~ l d on R 7. I t is clear t h a t ~ c o m m u t e s with t h e action a b o v e (since left and right multiplication commute). Hence, ~ descends to an action on t h e orbit space X.
An easy c o m p u t a t i o n shows t h a t the generic orbits of ff on S are two-dimensional, and t h a t S U ( 2 ) / X is diffeomorphic to a closed interval. The e n d p o i n t s of the interval correspond to the two singular orbits:
= {(x, 0 ) e n ' • Ilzll = 1} a n d = {(0, y)ea • llYil =
The family of m a p p i n g s i~: $ 3 ~ S 6, 0~cr ~ l, represents a curve in the orbit space between the two singular orbits, a n d therefore all the i s o m e t r y classes of orbits are re- presented b y this family. Hence, to find an orbit of m a x i m a l volume in S e we need o n l y find an orbit of m a x i m a l v o l u m e in this family.
L e t v(a) d e n o t e t h e v o l u m e of ia(S3). T h e n a c o m p u t a t i o n similar to the one in w 6 or a direct interpretation of formula (6.6) shows t h a t
v(a) = 27~2a(4 - 3~2).
This function reaches its m a x i m u m in [0, 1] at ~ = 2 / 3 . W e conclude t h a t
is a minimal submanifold of S e.
THE MIbTIMAL SURFACE SYSTEM 15 I t follows t h a t the cone over this submanifold is a minimal v a r i e t y in R 7. Therefore, we h a v e proved
THEOREM 7.1. The Lipschitz /unction /: R 4 ~ R 3 given by
/(x)=yl[xll
for x . O ,where ~] is the H o p / m a p (6.5), is a solution to the m i n i m a l sur/ace system (2.2).
I n particular, this shows that there exist Lipschitz solutions to (2.2) which are not o/class C 1 .
Note. This theorem has been verified b y direct computation.(1) Analogous examples can be constructed using the t t o p f maps
~ ' : $7--~ S 4
The corresponding volume functions are
y~4
v'(~) = ~ ~ ( 4 - 3~2) ~
and ~": S15~SS.
2:T~8 7
and v"(~) = ~ ~ (4 - 3o:2) 4.
Observe t h a t these submanifolds are all examples of compact minimal varieties in the sphere whose normal planes are at a constant acute angle with respect to a fixed plane in R N. By a result of de Giorgi this is not possible in eodimension one unless the sub- m~nifold is a totally geodesic subsphere. A result of Simons [36], refined by Reilly [34], states t h a t if M is a compact minimal submanifold of codimension-k in S N-1 with normM plane field v satisfying
V k-2
<~,~'0) > ~ 2
for some fixed k-plane ~0 in R N, then M is a totally geodesic subsphere. The examples above show t h a t there is a positive lower bound for the " b e s t " constant possible in this theorem.
In fact, for the H o p f maps ~], ~', and 7/" above, there are planes v0 such t h a t (v, v0) is constant and equal to 1/9, (1/8]/;/) and ( 7 ' . 2 -1'.3-5V7i5) respectively.
Remark. I t follows from the work of J. L. M. Barbosa ("An extrinsic rigidity theorem for minimal immersions from S ~ into S~, '' to appear) t h a t there are no non-parametric (1) We should like to thank the MACSYMA program at M.I.T. and Bill Gosper of the Stanford University Artifieal Intelligence Laboratory.
16 H. B, LAWSOlff, JR. AND R. OSSERMAN
m i n i m a l cones of d i m e n s i o n t h r e e o r less. Moreover, u s i n g t h i s f a c t one can p r o v e t h a t e v e r y L i p s c h i t z s o l u t i o n of t h e m i n i m a l surface s y s t e m in t h r e e (or fewer) i n d e p e n d e n t v a r i a b l e s is r e a l a n a l y t i c . Of course, b y t h e e x a m p l e s a b o v e such a s t a t e m e n t is false for t h e s y s t e m in f o u r o r m o r e v a r i a b l e s .
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Received August 23, 1976.
2 - 772904 ,,lcta mathematica 139. Iraprim6 le 14 Octobre 1977.
