We study SO(2)-invariant minimal and constant mean curvature surfaces in. R 3 endowed with a homogeneous Riemannian metric whose group of isometries ...
manuscripta math. 87,
1 -
12 (1995}
manuscripta mathematica Springer-Verlag1995
SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional h o m o g e n e o u s spaces
~*>
by
Renzo Caddeo, Paola Piu, Andrea Ratto R e c e i v e d 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
(1.1)
dx2 + dy2 + (dz + l ydx - xdy ~2 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 {x2 + 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 of 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' R1212 = 4m - ~
(1.2)
R1313 -
l2 4
R2323 -
l2 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,y,z) * ( x ,' y ,'z ) ' = (x + x', y + y', z + z' + xy' 2- x'y). 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:
The relevant O.D.E.
Section 3:
The prime integral.
Section 4:
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
2
3
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 }.
W e 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
~: (R 3, ds 2) --o (X,g)
(2.2)
is a Riemannian submersion over the set R 3 - ~, = Tc1(~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)
(2.5)
tip)_
1 +mp 2 1202 1+ 4
1202 p 1+ 4 V(p) [1 + m p 2 ]2
W e 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) and
f2(p) .z 2 + "02 = 1
4
(2.8)
R. Cacldeo et al.
d where " = d t '
Remark.
z,- z," "p]f(p)- z9 V'(p) f ( p ) - f ' ( p ) [ l +
2H=[p
V(p)
,~2] ~.
d ' - dp and f Vare given in (2.4), (2.5) respectively.
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. P r o o f o f Theorem 2.6. Step 1. Here we prove that (up to multiplication by an irrelevant positive constant) = V2(p) [f2(p) dz 2 + dp 2] ,
(2.9)
where f a n d V are given by (2.4) and (2.5) respectively. Since
is defined by (2.3),
we need to compute g and the volume function k'. We claim that (2.10)
(2.11)
1 g - [ 1 + rap2] 2 [f2(p) d z 2 + d p 2 ]
V =[l+mp
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 rc*(g) = (ds )tHor ,
(2.12)
where thor 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 m p 2]2 do2
4 [1 +
+ dz2+
+
[1 +
lP 2 [1 + rap2] d O d z .
1 m p 2]2 dP 2 +
S0(2)- invariant minimal and constant mean curvature surfaces a Clearly the vector - -
spans the vertical distribution. We set
(2.14)
a [1 + mP2] po3--
aO
V1 =
O
O0 ' az>dS2
c9
a
2l[1 + mp 2] O
ao
az
[4 + 12p 2] oo
'72 = ~-
lifo I
Using (2.13), we find that (2.15)
lie1 lids2 = 1
and
II W211as2
- -
1+
12 p 4
Then, by construction, the vectors
12 p v1
(2.16)
and
v2 = V2
1 +
4
form an orthonormal basis for the horizontal distribution. Passing to their dual forms we find (2.17)
=
[1
+ rap2]2
1 [1 + m p
2]2
[1 +
2 2 dz2
]
-
-
[dp 2 + f 2(p) dz 2]
Proof of (2.11). We have
21~
1+--~--
2/l"
dt = j[1 + m p 2] V(p) dt
(2.18)
[1 + mp ;] 0
from which (2.11) follows at once.
R. Caddeo et al.
6 Step 2. We set h = [dp 2 + f2(p)d z 2]
(2.19) and rewrite (2.9) as
= V 2(p)h.
(2.20)
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
2H = k(y)- -~ (log V) ,
(2.21)
where v is the unit normal to y with respect to the metric h, and
k(7) = < vh~ "7, v >h 9t
(2.22)
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' (log V) = z f -~
(2.24)
Next, a routine computation involving the Christoffel symbols of h gives '
(2.25)
+
3+
:':
2] O
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 j = V ( p ) f 2 ( p ) . z + 2 H S f ( u ) V(u) du o
(3.2)
is constant along Z i.e., dJ ~- -- 0.
(3.3) R e m a r k 3.4
At least until Section 4 below, inserting the explicit expressions (2.4),
(2.5) f o r f a n d 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
(3.5)
when m = 0
P
~f(u) V(u) du = f l +-i-~u 2 du = o o
t,ff-ml~
+ mp2)
w h e n m ;*0
This fact is very useful when one wants to use J to determine the qualitative behaviour of solutions (see Section 4 below).
P r o o 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)
(3.8)
2 H = - tip) [ "z" p L
2H = - -
f(p)
+ (2 f '(p)
[
tip)
+ V(p)) V'(P) 1
.
8
R. Caddco et al.
Next, we compute (along any solution
dJ dt
(3.9)
-
(z(t), p(t) ) )
2Hf(p) V(p)'p + ~V(p)fZ(p) +
+ V ,(p)f2(p) ~ .p+ 2 f '(p)f(p)V(p) z fl = f2(p) V(p) 2 H fi + "z" +
tip)
2 f'(p) z9 p9+ V'(p) 'z ] 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 dz 2
V'(p) [f2(p) + ( ~ f ] V(p)
-
f'(p)[f2(p) + 2 ( ~ f ]
2 H.~]~[ If2(p)+ (dp~213
And the prime integral J now takes the form
J-
(3.12)
p
V(p)f2(P)
(,tp~
+2H [flu) V(u) du o
fi~P) + Ldz3
dJ
(i.e., dzz -=0
4
wheneverp(z) is a solution of(3.1 I)).
The qualitative behaviour o f solutions
As we have already observed, any
SO(2)-invariant surface S 7 in (R 3, ds2) is completely (p,z)-plane (p >_0). The aim of this section is to
determined by its profile curve 7in the
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 4 0 . Case 1: H = 0.
In this case, the unique (see [ER], w 2, Chapter VI) solution starting is the trivial solution z =-- z o. Hence the
at any given boundary point (O, zo) of X
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) determined 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
(4.3)
dp = + dz -
,•fpZ ~-
1
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, S 0 ( 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 20.
1 Thus FI_I(O) = F H ( ~ ) = 0 ; FHis negative on (0, I ) ,
1
p = ~.
and attains its minimum at
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. Let H > 0 a n d 0 > c > FH(P*) be fixed. Then there exists a unique (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 profile curve ~. Moreover, z i s a wave-like p e r i o d i c curve which oscillates between the lines p -=m c a n d 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) Therefore J - c (4.7)
along y.
J _>Fn(P)
clearly implies along Z
m c
