manuscripta math. 87, 1 - 12 (1995} manuscripta mathematica

Springer-Verlag 1995

SO(2)-invariant minimal and constant mean curvature surfaces

in 3-dimensional homogeneous spaces ~*>

by

Renzo Caddeo, Paola Piu, Andrea Ratto

Received June 7, 1994

Abstract. We study SO(2)-invariant minimal and constant mean curvature surfaces in

R 3 endowed with a homogeneous Riemannian metric whose group of isometries has

dimension greater or equal to 4.

1 Introduction

We consider the following two-parameter family of Riemannian metrics on R 3

dx2 + dy2 + (dz + l ydx - xdy ~2 (1.1) ds2 = [1 + m(x 2 +y2)]2 2 1 + m(x 2 +y2)) l,m e R .

(However, note that, when m < 0 , formula (1.1) and all the calculations below are to be 1

understood as restricted to the set {x 2 + y2 < _ m } )"

These metrics have been known for a long time. They can first be found in the

classification of 3- dimensional homogeneous metrics given by L. Bianchi in 1897 (see

[BID; later, they appeared in the works of l~. Cartan (see [Ca], p.304) and G. Vranceanu

(*) Work partially supported by 40 % and 60 % Italian M.UR.S.T. funds.

2 R. Caddeo et al.

(see [Vr], p.354). Their geometric interest lies in the following fact: the family of metrics

(1.1) includes all 3-dimensional homogeneous metrics whose group o f isometries has

dimension d = 4 or 6, except for those of constant negative sectional curvature.

The sectional curvatures of the metrics (1.1) are given by (see [Pi])

3 12 ' l 2 l 2 (1.2) R1212 = 4m - ~ R1313 - 4 R2323 - 4 "

In particular, (1.1) provides the canonical lefl-invariant metrics of

(i) ff'L(2,R) , when l ~ 0 and m < 0,

(ii) the Heisenberg group/ /3 identified to (R 3, *), when l ;~ 0 and m = 0, where

* ' ' ' x', z' xy ' 2- x'y). (x ,y,z) ( x , y , z ) = (x + y + y', z + +

Very little is known about minimal and constant mean curvature surfaces in

(R z, ds2). Some examples of ruled minimal surfaces in H 3 were given in [BS]; and

minimal graphs over the xy-plane in H 3 were studied too, see [Be]. The aim of this paper

is to study SO(2)-invariant minimal and constant mean curvature surfaces in (R ~, ds2):

we work in the setting of cohomogeneity one equivariant differential geometry, along the

lines developped by Hsiang and his collaborators (see [Hs], [ER] and references

therein). Our main result is to show that, despite its rather complicated form, the relevant

O.D.E. (given in Theorem 2.6 below) admits a prime integral J. In the final section we

use J to describe the qualitative behaviour of solutions.

We hope that the present work will draw the attention of geometers on the metrics

(1.1) and stimulate further research on the Riemannian geometric aspects of these

metrics.

Our paper is divided in sections, as follows:

Section 2:

Section 3:

Section 4:

The relevant O.D.E.

The prime integral.

The qualitative behaviour of solutions.

Acknowledgements. This research was carried out while the third named author

Andrea Ratto was C.N.R. visiting professor at the University of Cagliari. He wishes to

thank these institutions for financial support and warm hospitality respectively.

SO(2)- invariant minimal and constant mean curvature surfaces 3

2 The relevant O.D.E.

The aim of this section is to prove Theorem 2.6 below. First, we note that S0(2) acts

isometrically on (R 3, ds 2) by rotations around the z-axis. Its associated orbit space can

be naturally identified with the set

(2.1) X = {(z,p) �9 R 2, p 2 0 }.

We denote by Z the z-axis and by X the interior of X: following [Hs], we can endow

X with a metric g in such a way that the canonical projection

(2.2) ~: (R 3, ds 2) --o (X,g)

is a Riemannian submersion over the set R 3 - ~, = Tc 1(~ O. There is an obvious bijective

correspondence between curves ~(t) = (z(t),p(t)) in X and SO(2)-invariant surfaces

S 7 = 7r -1(~) in (R 3, ds2). In particular, the mean curvature of S~, can be calculated in

terms of ~'.

Let ~" : X -o R _> 0 be the volume function (i.e., the function which associates to

a point x �9 X the volume of the orbit ~ l(x) ). We shall need to consider the metric

(2.3) ~ = ~2g on X.

For the sake of convenience, we set

(2.4) t i p ) _ 1 + m p 2

1202 1 + 4

1202 p 1 + 4

(2.5) V(p) - [1 + m p 2 ]2

We are now in the right position to state

Theorem 2.6 (The O.D.E.). An SO(2)-invariant surface Sy = z r 1(7) in (R3,ds 2)

has constant mean curvature H provided that its image curve ~(t) = (z(t),p(t)) in X

verifies

(2.7) f2(p) .z 2 + "02 = 1

and

4 R. Cacldeo et al.

�9 V'(p) (2.8) 2 H = [ p z,- z," "p]f(p)- z f ( p ) - f ' ( p ) [ l + ,~2] ~. V(p)

d d where " = d t ' ' - dp and f Vare given in (2.4), (2.5) respectively.

Remark . Condition (2.7) is equivalent to the parameter t being the arc length with

respect to a suitable metric (see Step 2 below). This condition is useful to simplify

calculations.

Proo f o f Theorem 2.6.

Step 1. Here we prove that (up to multiplication by an irrelevant positive constant)

(2.9) = V2(p) [f2(p) dz 2 + dp 2] ,

where f a n d V are given by (2.4) and (2.5) respectively. Since

we need to compute g and the volume function k'. We claim that

1 (2.10) g - [ 1 + rap2] 2 [f2(p) d z 2 + d p 2 ]

is defined by (2.3),

(2.11) V = [ l + m p 2]V(p) (up to multiplication by a positive constant).

Now (2.9) follows immediately from (2.3), (2.10) and (2.11). So, in order to complete

our Step 1, it only remains to verify (2.10) and (2.11).

Proof o f (2.10). The metric g is defined through the condition

2 (2.12) rc*(g) = (ds )tHo r ,

where tho r denotes restriction to the horizontal distribution (with respect to the projection

•). It is convenient to operate in cylindrical coordinates x = p cosO , y = p sinO , z = z ,

with respect to which the metric ds 2 of (1.1) takes the form

(2.13) ds 2 4 p 2 + l 2p4 1 - 4 [1 + m p 2]2 do2 + [1 + m p 2]2 dP 2 +

+ dz2+ lP 2 [1 + rap2] d O d z .

S0(2)- invariant minimal and constant mean curvature surfaces

a Clearly the vector - -

aO spans the vertical distribution. We set

(2.14) a

V 1 = [1 + mP2] po3--

'7 2 O O0 ' az>dS2 c9 a 2l[1 + mp 2] O

= ~ - l i f o I ao az [4 + 12p 2] oo

Using (2.13), we find that

(2.15) lie 1 lids 2 = 1 and II W211 as 2 - -

12 p 1+ 4

Then, by construction, the vectors

12 p (2.16) v 1 and v2 = V2 1 + 4

form an orthonormal basis for the horizontal distribution. Passing to their dual forms we

find

(2.17) = 2 2 d z 2 - [ 1 + rap2] 2 [1 + ] -

1 [dp 2 + f 2(p) dz 2] [1 + m p 2]2

Proof of (2.11). We have

(2.18)

21~

0

1 + - - ~ - -

[1 + mp ;]

2/l"

dt = j[1 + m p 2] V(p) dt

from which (2.11) follows at once.

6

Step 2. We set

(2.19)

and rewrite (2.9) as

(2.20)

h = [dp 2 + f2(p)d z 2]

= V 2(p)h.

R. Caddeo et al.

Now we follow [Hs] or [ER] p.61: we suppose that the parameter t is the arc length with

respect to the metric h, a fact which is equivalent to (2.7). Thus we have

3 (2.21) 2H = k(y)- -~ (log V) ,

where v is the unit normal to y with respect to the metric h, and

(2.22) k(7) = < vh~ "7, v >h 9t

is the geodesic curvature of y with respect to h. By way of summary, it remains to show

that the explicit form of (2.21) is given by (2.8). Indeed, we have

(2.23) v = - 7 - ~ + zf~pp ,

from which we deduce that

�9 V ' (2.24) (log V) = z f -~

Next, a routine computation involving the Christoffel symbols of h gives

' 3+ 2] O (2.25) + : ' :

0t

Thus, using (2.25) and (2.23) into (2.22) we obtain

(2.26) k(),)= - "pf ['z + 7 ~ "p] + z f [ p - f ' f "z2]

Finally, replacing (2.24) and (2.26) into (2.21), rearranging the order of terms and using

(2.7) we obtain (2.8), so that the proof of Theorem 2.6 is completed.

S0(2)- invariant minimal and constant mean curvature surfaces

3 The prime integral

We show that the O.D.E. (2.7), (2.8) admits a prime integral J. Indeed, we have

Theorem 3.1. Let ~(t) = (z(t),p(t)) be a solution of(2. 7), (2.8). Then the quantity

P (3.2) j = V(p) f2 (p ) . z + 2 H S f (u ) V(u) du

o

is constant along Z i.e.,

dJ (3.3) ~- -- 0.

Remark 3.4 At least until Section 4 below, inserting the explicit expressions (2.4),

(2.5) f o r f and V into (2.8) or (3.2) is not particularly illuminating, so we do not do it.

However, we point out that the special form of f and V always enables us to compute

explicitly the integral which appears in the definition of J. Indeed, from (2.4) and (2.5)

we easily deduce

p P when m = 0

(3.5) ~f(u) V(u) du = f l +-i-~u 2 du =

o o t, f f-ml~ + m p 2 ) when m ;*0

This fact is very useful when one wants to use J to determine the qualitative behaviour of

solutions (see Section 4 below).

Proo f o f Theorem 3.1. Since (2.7) holds, we have (taking derivatives with respect

to t):

(3.6) ~ "~ + ~ ~ f 2 ( p ) + "zef(p)f,(p)'p = 0 .

Using (3.6) and (2.8), it is easy to deduce the following (useful) forms of (2.8):

(3.7) 2 H = - tip) [ "z"

L p

+ (2 f '(p) [ t ip ) + V(p))

(3.8) 2 H = - -

f ( p )

V ' (P) 1 .

8 R. Caddco et al.

Next, we compute (along any solution (z(t), p(t) ) )

dJ 2Hf(p) V(p)'p + ~V(p)fZ(p) + (3.9) dt -

+ V ,(p)f2(p) ~ .p+ 2 f '(p)f(p)V(p) z fl

2 f ' (p) �9 �9 V'(p) ] = f2(p) V(p) 2 H fi + "z" + z p + 'z tip) f(p) V(p)

Now (3.3) follows immediately from (3.7) and the theorem is proved.

R e m a r k 3.10 The solutions which are graphs over the z-axis, i.e., those of the form

p = p(z), have special interest. In this case a calculation shows that conditions (2.7),

(2.8) become

(3.11) d2p V'(p) [f2(p) + ( ~ f ] f ' ( p ) [ f2 (p ) + 2 ( ~ f ] dz 2 V(p) -

2 H.~]~[ I f2 (p )+ (dp~213

And the prime integral J now takes the form

p (3.12) J - V(p)f2(P) +2H [flu) V(u) du

(,tp~ o fi~P) + Ldz3

dJ (i.e., dzz -=0 wheneverp(z) is a solution of(3.1 I)).

4 The qualitative behaviour o f solutions

As we have already observed, any SO(2)-invariant surface S 7 in (R 3, ds 2) is completely

determined by its profile curve 7in the (p,z)-plane (p >_ 0). The aim of this section is to

show how the prime integral J of Section 3 can be used to deduce the qualitative

behaviour of solution curves 7. Since this amounts to a case by case analysis of similar

instances, we shall present in detail only some significant examples, leaving the

remaining ones to the interested reader. In particular, in order to simplify the exposition,

SO(2)- invariant minimal and constant mean curvature surfaces 9

f r o m now on we restrict our attention to the Heisenberg metric, i.e., we f i x m = 0 and l

= 1. It is convenient to separate 2 cases, according to whether H = 0 or H 40 .

Case 1: H = 0. In this case, the unique (see [ER], w 2, Chapter VI) solution starting

at any given boundary point (O, zo) of X is the trivial solution z =-- z o. Hence the

uniqueness principle for the Cauchy problem implies at once that all the solutions to

(2.7),(2.8) (with H = 0) are graphs /9 = p(z ) over z-axis. Therefore, according to

Remark 3.10, we can restrict our attention to (3.11) (with H = 0 there). From (3.12),

(2.4), (2.5) and (3.5) it is easy to see (recalling that m = 0 = H, l = 1 !) that the condition

J = c along a solution takes the form

" ~ / 1 + ~ -

Then it is not difficult to estimate that the solution pc(Z) de termined by the initial

condition p(O) = c > 0 is an even function o f z , strictly increasing f o r z > O, which tends 2

asymptotically to a line o f the f o r m p(z) = c z + constant as z --~ ~o. Since the problem

is invariant by translations in the z-direction, we also deduce that all solutions are o f the

type

(4.2) p(z) = pcl(Z + c 2)

with c I > 0 and c 2 ~ R . It is also worthwhile to compare (4.1) with the corresponding

condition which arises in the Euclidean case, namely

,•fpZ dp = + 1 (4.3) dz - ~ -

where the required solutions are the usual catenoids p(z) = c cosh( z + a), a ~ R (For

instance, one finds that pc(z) ~- c cosh z for all z E R , with equality only at z = 0 ).

Case 2: H ;~ O. In Proposition 4.5 below we obtain surfaces in H 3 which resemble the

classical Delaunay surfaces of the Euclidean case. As a limit case of Proposition 4 .5 , we

shall find constant mean curvature, embedded, S0(2) - invariant 2-spheres in H 3 .

Without loss of generality, we assume H > 0. Our analysis requires an auxiliary

function F H defined as follows:

10 R. Caddeo et al.

(4.4) F~t (p) = H p2 . p p 2 0 .

1 Thus FI_I(O) = F H ( ~ ) = 0 ; FHis negative on (0, I ) and attains its minimum at

, 1 p = ~ . Moreover, given 0 > c > F~t(p*) , there exist exactly 2 points m c < M c such

that F H (mc) = FIz (Mc) = c. We have

P r o p o s i t i o n 4.5. Le t H > 0 and 0 > c > FH(P*) be f ixed . Then there exists a un ique

(up to translation in the z-direction) SO( 2 )-invariant surface S}, in H 3 with constant mean

curvature H, such that J ~ c along its profi le curve ~. Moreover, z i s a wave-l ike per iod ic

curve which osci l lates be tween the l ines p -=m c and p - M c .

Proof . First we observe that for m = O, l = 1, condition (2.7) implies

(4.5) _>- 1 along Z

1 + ~ -

Inserting (4.5) in the definition (3.2) of J we deduce that

(4.6) J _>Fn(P) along y.

Therefore J - c clearly implies

(4.7) m c <_ p(t) _< M c along Z

Next, we fix z o e R and consider the unique solution of (2.8) determined by the

following initial conditions

z(O) = z o , p(O) = M c

(4.8) 4 M c 2 z(0) = - 1 + ~ - , p(0) = 0

The rest of the proof follows by the arguments used in [ER] p. 84, and so we only

sketch it. (However, we take this opportunity to point out that the statement contained in

the lines 3 and 4 of p. 85 of [ER] is false and should be deleted without further

modification). First, we observe that P2 - - M c is no t a solution of (2.8), and so, as t

increases from 0, y must head towards the interior of the strip [m C, Mc]. Next, one

proves that ~)(t) < 0 until t reaches a value tj at which

S0(2)- invariant minimal and constant mean curvature surfaces 11

2 m c

(4.9) p(t / ) = m "p(t z) = 0 and "z(t I) = - I + ~ .

It lbllows easily froln (4.9) that we can continue y by reflection across the line z -~z(t~),

and iterating this procedure we conclude that y is a wave-like periodic curve (of period

2t I ) , as required to end the proof.

The two limit values c = FH(p*) and c = 0 are interest ing. The fo rmer 1

corresponds to the solution p -=p* = ~ . The associated cylinder in H 3 has zero

Gaussian curvature and is convex. It would be nice to know whether these cylinders arc

(up to isomctry) the only conlpletc, convex, COllSt;.ln[ 111121.111 cttt-v;tturc SLu'fac~2S ill 11.~.

In the case c = O, we f ind constant mean curvature embedded 2-spheres: more precisely,

we can write the profile curve 7 as a graph over the z-axis, described by a function p =

p(z), a <-z _<b tbr some a < b. Using (3.12) and the explicit expressions f o r f a n d V, we

find that the condition J ==0 becomes

(dp ' 1 'o- (4.10) ~,dz) . . . . H

1 + 4

Comparing (4.10) with the following condition, verified by the profile of Euclidean

sphercs,

(d__p_) 2 [1 - p 2 H 2 ] (4.11) \ d z ) - H 2 p 2

it is not difficult to estimate that the shape of the profile of a constant mean curvature,

S 0 ( 2 ) -invariant sphere in H~ is qualitatively as the dotted curve in the Figure below

(the other curve there represents a half-circle).

12 R. Caddeo et a].

References

[Be] M. Bekkar, Exemples de surfaces minimales dans l'espace de Heisenberg,

Rend. Sere. Fac. Sci. M.F.N. Cagliari, vol. 61, 2, (1991) pp. 123-130

[BS] M. Bekkar and T. Sari, Surfaces minimales regl~es dans l'espace de

Heisenberg H 3, Rend. Sere. Mat. Univ. e Politec. Torino, vol. 50, 3, (1992)

pp.243-254

[Bi] L. Bianchl , Lezioni sulla teoria dei gruppi continui e f init i di

trasformazioni, Ed. Zanichelli, Bologna (I 928)

[Ca] I~. Cartan, Lemons sur la g~om~trie des espaces de Riemann, Gauthier

Villars, Paris (1946)

[ER] J. Eeils and A. Ratto, Harmonic maps and minimal immersions with

symmetries, Annals of Mathematics Studies, Princeton University Press 130,

(1993) pp. 1-228

[Hs] W.Y. Hsiang, Generalized rotational hypersurfaces of constant mean

curvature in the Euclidean spaces I, J. Diff. Geom. 17 (1982), pp. 337-356

[Pi] P. Piu, Sur certains types de distributions non-integrables totalement

g~od~siques, Th~se de Doctorat, Univ. de Haute Alsace, (1988)

[Vr] G. Vranceann, Legons de g~omdtrie diff~rentielle, Ed. Acad. Rep. Pop.

Roum., vol I, Bucarest (1957)

Renzo Caddeo and Paola Piu, Dipartimento di Matematica,

via Ospedale 72, 1-09124 CAGLIARI, Italy

Andrea Ratto, Ddpartement de Math6matiques - U.B.O.,

6, Av. Le Gorgeu - BP 452, F-29275 BREST, France