A complete minimal surface in R3 between two parallel planes

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Annals of Mathematics

A Complete Minimal Surface in R3 Between two Parallel Planes Author(s): Luquesio P. de M. Jorge and Frederico Xavier

Reviewed work(s):

Source: Annals of Mathematics, Second Series, Vol. 112, No. 1 (Jul., 1980), pp. 203-206 Published by: Annals of Mathematics

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A complete

minimal

surface in R3

between two parallel planes

By LuQufEsio P. DE M. JORGE* and FREDERICO XAVIER**

E. Calabi has asked if it is possible to have a complete minimal surface in RX entirely contained in a half-space. We answer his question in the affir- mative. In fact, we prove considerably more, namely, we show how to produce complete minimal surfaces contained in slabs of R3. The proof was motivated by Remmert's ingenious idea of using Runge's theorem to exhibit proper analytic embeddings of the unit disc D c( C in C3, as explained in [11,

page 96.

It should be pointed out that M. Miranda 121 has shown that a surface that minimizes area globally and lies in a half-space of RX is a plane. Here, however, we deal with surfaces that minimize area only locally. We also observe that B. Lawson [51 has given examples of complete surfaces of con- stant mean curvature which lie between two parallel planes.

We take the oportunity to call attention to _[4jwhere a related conjec- ture is discussed.

The Example

LEMMA 1. (Weierstrass representation of minimal surfaces). Let

f,

g: D SAC be holomorphic and set

01

=-f(lg2) 02 = f(l + g2) 03= fg

2 2

If f never vanishes then the function x(xl, X2, X3): D R3, where Xk Re

will define a (regular) minimal surface in R3 whose element of length is given by ds = X _{I dz I where}

X If I + IgV12) 2

LEMMA _{2. Let {DJ} be a sequence of closed discs centered at the origin,} Dn ci Dn+1, U Dn = D. Let Kn ci _{D. be a compact set such that}

Kn

_nfDn

l

= 5

0003-486X/80/0112-1/0203/004 $ 09.20/1

(? 1980 by Princeton University (Mathematics Department) For copying information, see inside back cover.

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204 L. P. DE M. JORGE AND F. XAVIER

and D\K,, is connected. Let f be a holomorphic function defined in a neigh- borhood of the union of the K,. Then it is possible to approximate f uni- formly on that union by holomorphic functions defined in D.

For the proof of Lemma 1, see 3:1. Lemma 2 is precisely exercise G of [1], page 96; there the reader will also find a sketch of its proof using Runge's theorem.

THEOREM. There are nonfiat complete minimal surfaces of RK entirely

contained in a slab.

Proof. Consider the picture below.

Kn

As indicated, K,, is the compact region formed by deleting a piece from an annulus. Let

r,

be the difference between the outer and inner radii of the annulus. For the next compact set, that is, K,+,, we delete a piece from another annulus (disjoint from the first) but from the "other side", and so on. Let

E =Uneven Kn and

0

= U,,odd

K,,.

We shall say that a path a crosses K,, if it intercepts both the inner and outer arcs of the annulus from which K,, is obtained. A little reflection shows that

_{KJ}

satisfies the following:

(*) Any divergent path in D of finite Euclidean length will cross all but a finite number of the K,, in E or all but a finite number of the K,, in 0.

Note that if K, is chosen so that its "opening" is to the right of 0 then the segment

{(x, 0) ; - < x <

2

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A COMPLETE MINIMAL SURFACE IN R3 205 Let {C"} be a sequence of positive numbers, to be specified later. By

Lemma 2 there is a holomorphic function h on D such that

_I

h - _c,[_I_<1 on

K,,. Let eh be the function g appearing in Lemma 1 and set

f-

1/g. Con-

sider the minimal surface determined by

f

and g, as in Lemma 1. An im- mediate consequence is that x3 is bounded; that is, our minimal surface M is contained in a slab. We will show that {Ca,} can be chosen so as to make

M complete. Let a be a divergent path in D, which may be supposed to have Euclidean speed one. We shall distinguish two cases.

1) Suppose that a has infinite Euclidean length, i.e., a: [0, oo) D. Since

1Ig + > 1,

2 \ _g

it follows that

1 (a) = _{X(a(t)) dt}=o

0

2) Suppose now that a has finite Euclidean length, i.e., a: [0, b)

_D,

b < cO. Consider the first alternative given in * and let m be an integer such that a crosses every K& C E with n > m. From g = eh - ecne hcn we

have Ig

I

> eCn1 on K,,. Let

Jo = It e [0, b) I a(t) e K,}

-

Then

21(at) >- I g(at(t))|

d t

> En~,n vnS g(at(t)) I

d t

?Enmnevenecnl dt _>-n~ m,n even recnl

(recall that t is the Euclidean arc-length of a).

Similarly, if the other alternative in * holds we have

1(a) ? En!k,nodd r,,ecn1 .

Therefore, if cn is chosen to grow fast enough, the curve a will have infinite

length in M. In fact, it suffices to take c, = -log r-,,. UNIVERSIDADE FEDERAL DO CEARA, FORTALEZA-CE-BRAZIL UNIVERSIDADE FEDERAL DE PERNAMBUCO, RECIFE-PE-BRAZIL

REFERENCES

[ 1 ] K. HOFFMAN, Banach Spaces of Analytic Functions, Prentice-Hall, Inc., Englewood Cliffs,

N.J. (1962).

[21 M. MIRANDA, Frontiere minimali con ostacoli, Annali Dell UniversitA di Ferrara, Vol. XVI, N2 2 (1971), 29-37.

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206 L. P. DE M. JORGE AND F. XAVIER

[ 4] L. P. M. JORGE and F. XAVIER, _{On the existence of complete}_{bounded minimal surfaces}

in RN, to appear in Bol. Soc. Bras. de Mat., Vol. 10, N' 2 (1979).