proceedings of the
american mathematical society
Volume 123, Number 9, September 1995
COMPLETE HYPERSURFACES
WITH CONSTANTMEAN CURVATURE
AND NON-NEGATIVESECTIONAL CURVATURES
HU ZE-JUN
(Communicated by Christopher B. Croke)
Abstract.
We classify the complete and non-negatively curved hypersurfaces
of constant mean curvature in spaces of constant sectional curvature.
1. Introduction
Let M
(c) be an (zz+1 )-dimensional space of constant sectional curvature
c. When c < 0, Mn+\c) = Hn+X(c); when c = 0, M"+\c) = R"+x ; when
c > 0, M
(c) = Sn+X(c), respectively. Let M" be an zz-dimensional hyper-
surface with constant mean curvature H in M
(c). Let S denote the square
of the length of the second fundamental form. The main purpose of this paper
is to give a characterization of non-negatively curved hypersurfaces of M
(c)
by the relationship between S and H, this can be compared with the results
obtained by Nomizu and Smyth [3], Okumura [4], Goldberg [1], Hasanis [2]
and Smyth [6].
Theorem. Let M" be a complete non-negatively curved hypersurface of M
with constant mean curvature H. Then M" is totally umbilical or
_
zz3//2
. n(n-2k),TT.
/ lTTl
SUP5= nC+ 2k(n^T) ± Jkl^Y)^V"2"2
for some k= 1,2,
77"
(c)
77"
+ 4*(" - Ve>
... , n — 1, when c > 0 ; or
supS = zzc-f -^-—
- ^—-f
\H\Jn2H2 + 4(n - l)c,
2(zz- 1) 2(zz- 1)
v
when c < 0.
In particular, if M" is connected and S = constant (when c > 0, S =
constant may be replaced by the condition that M" is compact), then in the
second case we have the following:
(I) When o0,
M" =Sk(cx) x Sn~k(c2), for some k = 1, 2,...,
where cx> 0, c2 > 0 and l/cx + I /c2 = l/c.
n - I,
Received by the editors October 21, 1992 and, in revised form, February 3, 1994.
1991 Mathematics Subject Classification. Primary 53C40; Secondary 53C20.
Key words and phrases. Hypersurfaces of constant mean curvature, sectional curvature, totally
umbilical.
© 1995 American Mathematical Society
2835
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2836
HU ZE-JUN
(2) When c = 0, M" = Rk x Sn~k(cx) for some k = 1, 2, ... , zz- 1,
where cx > 0.
(3) When c < 0, M" = Hx(cx) x Sn~x(c2), where cx < 0, c2 > 0 and
l/cx + l/c2 = l/c.
From the theorem, we easily see the following
Corollary. Let Mn be a compact non-negatively curved hypersurface of M
(c)
(c < 0) with constant mean curvature H. Then Mn is totally umbilical and
has positive sectional curvature H2 + c.
Remark. The main theorem was partially proved by Nomizu and Smyth in [3]
whose results were extended to arbitrary codimension by Smyth [6] and later by
Yau in [7] under the condition that M" is compact.
2. Preliminaries
Let M" be a hypersurface of M" (c) and let ex, ... , en, en+x be a local
field of orthonormal frames in M
(c), such that, restricted to M" , the vector
field en+i is normal to M" . Then the second fundamental form B and the
mean curvature H for M" can be written as
(2.1)
B = Y,hij(OiO)je„+x, H = (l/iOj^A«.
i,j
i
The Gauss equation for M" is
(2.2)
i? = zz(zz-l)c + zz2//2-6\
where S - trB2 —YV ■/z2- and R denotes the scalar curvature of M" .
We denote by A the Laplacian relative to the induced metric on M".
H = constant, then ([3])
(2.3)
If
(1/2)A5 = |V5|2 - S2 + ncS - n2cH2 + nHtrB3.
For any point p in M" , we can choose a local frame field ex, e2, ... , e„ so
that the matrix (h¡j) is diagonalized at that point, say, h¡j = Xiôy . Then (2.3)
can be rewritten as ([3])
(2.4)
(1/2)A5 = |VZ?|2+ £(A¿ - Xj)2KtJ,
i<j
where K¡j —c + X¡Xj is the sectional curvature of the plane section spanned by
e, and e¡.
Lemma (see Omori [5] and Yau [8]). Let Mn be an n-dimensional complete
Riemannian manifold whose Ricci curvature is bounded from below. Let F be
a C2-function bounded from above on M" . Then for any e > 0, there exists a
point p in M" such that
supF-e<F(p),
\gradF\(p) < e,
3. Standard
AF(p)<e.
models
This section is concerned with some standard models of complete nonnegatively curved hypersurfaces with constant mean curvature of the space form
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COMPLETE HYPERSURFACESWITH CONSTANT MEAN CURVATURE
2837
M
(c). In particular, we only consider non-totally umbilical cases, and the
length of the second fundamental form of such hypersurfaces are calculated.
First, we consider a class of hypersurfaces Rk x S"~k(cx) of R"+x , where
k = 1, 2, ... , zz - 1 . The number of distinct principal curvatures of each
hypersurface is exactly two, say 0 and yfcx, with multiplicities k and zzk, respectively. The sectional curvatures of a plane spanned by two principal
directions are 0 and cx, respectively. It is easily seen that H and S are
constant, and they satisfy, for Rk x S"-k(cx) in Rn+X, S = n2H2/(n - k).
We next consider the case c > 0. Let
Sk(cx) = {(xx,...,xk+x)£Rk+l;x2
Sn~k(c2) = {(yx,...,
+ --- + x2+x = l/cx},
yn-k+i) 6 R"-M
Sn+X(c) = {(xx,...,
xk+x, yi,...,
; y\ + ■-■+ y„2_fc+)= l/c2},
y„-k+i) £ Rn+2;
xi+--- + 4+i+y¡
+ ---+y2n-k+i= ! M>
where l/c, + l/c2 = l/c, k = 1, 2, ... , zz- 1. Then Sk(cx) x Sn~k(c2) isa
family of hypersurfaces in Sn+' (c). The number of distinct principal curvatures
of such hypersurfaces in this family is exactly two, and they are constant. One
principal curvature is equal to ±y/cx - c with multiplicity k, and the other
is equal to +y/c2 - c with multiplicity n - k . The sectional curvatures of a
plane spanned by two principal directions are cx, c2 and 0, respectively. For
Sk(cx) x Sn~k(c2) in Sn+X(c) we can easily show that
_
zz3//2
n(n-2k).TT.
r~777,
~
S = nC+ Wn^k) ± Wn^F)^HWn2H2 + ^
77"
- k)c
where the plus (resp. minus) sign is taken if ky/cx - c > (n - k)y/c2 - c (resp.
ky/cx - c < (n - k)y/c2 - c).
Finally, we consider the case c < 0. Let
Hn+X(c) = {(x0, x,, ... , xn+x) £ Rnx+2; -x¡
Define a family of hypersurfaces
Hk(cx) x S"-k(c2) = {(xo, xx,...,
+ x2 + ■■■+ x2n+x= l/c}.
Hk(cx) x S"~k(c2) in Hn+X(c) by
xn+x) £ Rnx+2;
- xl + x\ + ■■■+ X¡ = 1/Ci , x¡+x + ■■■+ x2+x = l/c2},
where cx < 0, c2 > 0, l/cx + l/c2 = l/c and k = 1, 2,..., n - 1. The
number of distinct principal curvatures of such hypersurfaces in this family is
exactly two and they are constant. One principal curvature is equal to y/Cx- c
with multiplicity k and the other is equal to y/c2 - c with multiplicity n-k .
The sectional curvatures are 0, c2 and Ci for 1 < k < n ; they are 0 and c2
for k = 1. For Hk(cx) x S"rk(c2) in Hn+X(c), we can also easily show that
0
zz3//2
, n(n-2k).TT.
i -TT~
7T~
7~
S = nC+ Wn-^k) ± Wn^lT) 1^1V"2//2+ ^n - k)c>
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2838
HU ZE-JUN
where the plus (resp. minus) sign is taken if ky/cx - c > (n - k)y/c2 - c (resp.
kyjcx - c < (n - k)y/c2 - c).
4. Proof of Theorem
Since Kij = c + X¡Xj> 0 for any distinct indices i and j , (2.4) means that
(4.1)
(1/2)A5= \VB\2+ Y,(h-Xjf(c + XiXj)> 0.
i<j
On the other hand, K¡j > 0 implies that the Ricci curvature of M" is
bounded from below by zero. From (2.2) we have that S is bounded from
above by a constant zz(zz- l)c + n2H2. Then we can apply the lemma to the
function S. Then we get {pm} in Mn suchthat
(4.2)
lim S(pm) = supS,
m—oa
lim AS(pm) < 0.
m—>oo
This implies that (4.1) and (4.2) give rise to
(4.3)
(c + XiXJ)(Xl-XJ)2(pm)^0
(zzz^oc),
for any distinct indices i and j .
Now, since S = Y^A2 is bounded, any principal curvature Xj is bounded
and hence so is any sequence {Xj(pm)}. Then there exists a subsequence {pm'}
of {Pm} such that
o
(4.4)
o
Xj(pmi) -» Xj
(zzz'—>oo) for some A, and any y.
o
In fact, since a sequence {Xx(pm)} is bounded, it converges to some Xx by
taking a subsequence {pm,} if necessary. For the point sequence {pm¡}, a
sequence {X2(pm¡)} is also bounded and hence there is a subsequence {pmi} of
o
{Pmx} suchthat {X2(pm2)} converges to some X2 as m2 tends to infinity. Thus
we can inductively show that there exists a point sequence {pm>} of {pm} such
that the property (4.4) holds. By (4.3) and (4.4) we get
(4.5)
(c + Xl°Xj)(Xi-Xj)1= 0,
for any distinct indices i and j. By a simple algebraic calculation it is easily
o
seen that the number of distinct limits in {X¡} is at most two
o
Case 1. If all limits X¡ coincide with each other, because S - nH2 = YJX2(l/zz)(YJA,)2 = (1/zz) Y_V-(A/- Xj)2, it follows from the above property that
limm'_00(5' - nH2)(pmi) = 0. Combining this with (4.2), we get supS — nH2.
Hence the function 5 becomes a constant zz//2. Accordingly, the hypersurface
M" is totally umbilical.
o
Case 2. If {A,} has exactly two distinct elements, without loss of generality,
we may set
o
o
o
xx = • • • = Xk = X,
for some fc = l,2,...,n—
(4.6)
o
Xk+X= • • • = X„ = p,
1. From (4.5) we have
Xp = -c.
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X<p,
COMPLETE HYPERSURFACESWITH CONSTANTMEAN CURVATURE
2839
Now, we only consider the case c < 0. As for the cases c > 0 and c = 0,
the proof is simpler and similar to each other, we omit them.
Without loss of generality, we may assume that H > 0. Then two limits X
and p are positive. In this case we define a negative number Cx and a positive
number c2 by X2 = cx - c and p? = c2 - c, respectively, so these numbers
satisfy
(4.7)
c < Cx< 0,
c2 > 0,
{ 3fc~l~:rcJ(C2- c) = -c,
i.e.. I/e, + l/c2 = l/c.
By H = (l/n)Y,Xi(pm<) = constant, we have
(4.8)
nH = kX + (n - k)p = ky/cx - c + (n - k)y/c2 - c.
At the same time we have
supS=
(4.9)
lim y^X2(pmi) - kX2 + (n - k)p2
m'-+¡x,*—'
= k(cx -c) + (n- k)(c2 - c).
On the other hand, by using (4.7) and (4.8) we can easily see that
zz3//2
, n(n-2k).TT.
/ .T„
nc + Ö77-m ± 077-n\H\\Jn2H2
2k(n - k) 2k(n - k) '»
77"
77"
+ 4k(n - k)c
= k(cx -c) + (n-k)(c2-c),
where the plus (resp. minus) sign is taken if ky/cx - c > (n - k)y/c2 - c (resp.
kyjcx - c <(n - k)y/c2 - c).
Thus, from (4.9) we have
zz3//2
™PS = nC+ 2k(n-k)
, n(n-2k).TT.
f~7~.
± Wn-Y)^V"2H2
When k > 1, the sectional curvature
~
77"
+ 4k{n~k)C-
Kx2(pmi) — (c + XxX2)(pmi) —» c +
m'—>oo
X2 = cx < 0, so from the assumption we have k = 1. Then ky/cx - c <
(n - k)y/c2 - c. Therefore we have at last
0
zz3//2
supS = nc+2{n_l)
n(n-2).TT.
77777
~
77"
- 2y{n_l')\H\yJn2m + 4(zz- l)c.
This completes the proof of the first part of the theorem.
In particular, if S = constant and M" is connected, then (2.4) says that all
the principal curvatures are constant and that they satisfy
(4.10)
(c + XiXj)(Xi-Xj)2= 0,
for any distinct indices i and j.
{Xj} is at most two.
Hence the number of distinct elements in
If Xx= ■■■= X„, then M" is totally umbilical.
If {X¡} has exactly two distinct elements, without loss of generality we may
assume that Xx = ■■■= Xk = X, Xk+X- ■■■ = X„ = p, X < p, for some
zc=l,2,...,zz-l.
From (4.10) we have
(4.11)
Xp = -c.
For c > 0, the theorem has been proved by Nomizu and Smyth [3], so we
just consider the case c < 0. Without loss of generality we may assume that
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2840
HU ZE-JUN
H > 0. Then, similar to the first part, we get X = y/cx - c, p = yfc2 - c and
k = 1, where cx, c2 satisfy (4.7). Thus the second fundamental form of M"
in Hn+X(c) is given by
(hij) = diag(y/CX -C,
y/C2-C,
... , y/C2 - C).
Then, by using the method similar to that of [3] and combining with Section
3, we can show that M" is isometric to Hx(cx) x S"~x(c2). Q.E.D.
Acknowledgments
I would like to express my deep gratitude to Professor Christopher B. Croke
and the referee for their many helpful suggestions and their corrections on this
paper.
References
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Amer. Math. Soc. 79 (1980), 268-270.
2. T. Hasanis, Characterization of totally umbilical hypersurfaces, Proc. Amer. Math. Soc. 81
(1981), 447-450.
3. K. Nomizu and B. Smyth, A formula of Simons' type and hypersurfaces with constant mean
curvature, J. Differential Geom. 3 (1969), 367-377.
4. M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor,
Amer. J. Math. 96 (1974), 207-213.
5. H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967),
205-214.
6. B. Smyth, Submanifolds of constant mean curvature, Math. Ann. 205 (1973), 265-280.
7. S. T. Yau, Submanifolds with constant mean curvature, Amer. J. Math. 97 (1975), 76-100.
8. _,
Harmonic functions on complete Riemannian manifolds, Comm. Pure. Appl. Math.
28(1975), 201-208.
Department of Mathematics, Zhengzhou University, Zhengzhou, 450052, Henan, People's Republic of China
Current address: Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan,
People's Republic of China
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