On the Index of Constant Mean Curvature Hypersurfaces

0.12 MB·English

by  E. Colberg

ON THE INDEX OF CONSTANT MEAN CURVATURE
HYPERSURFACES
9
0 E.COLBERG,A.M.DEJESUS,K.KINNEBERG,G.SILVANETO
0
2
Abstract. In1968,Simons[9]introducedtheconceptofindexforhypersur-
n facesimmersedintotheEuclideansphereSn+1. Intuitively,theindexmeasures
a
the number of independent directions in which a given hypersurface fails to
J
minimizearea. Theearliestresultsregardingtheindexfocusedonthecaseof

8 minimalhypersurfaces. Manysuchresultsestablishedlowerboundsforthein-
2 dex. Morerecently,however,mathematicianshavegeneralizedtheseresultsto
hypersurfaceswithconstantmeancurvature. Inthispaper,weconsiderhyper-

] surfaces ofconstant meancurvature immersedintothesphereand givelower
G
bounds fortheindexundernew assumptionsabouttheimmersedmanifold.
D
.
h
t
a 1. Introduction
m
The index of minimal hypersurfaces immersed into the Euclidean sphere Sn+1
[
appeared in the seminal work of Simons [9]. Intuitively, the index measures the

1 number ofindependent directions inwhichthe hypersurfacefails to minimize area.
v
Inthispaper,Simonsproved,amongmanyresults,thattheindexofsuchhypersur-
8
facesis greaterthanor equalto one,with equalityonly attotallygeodesicspheres.
9
3 Later,Urbano[10]establishedthelowerboundforthecaseofminimalsurfacesinto
4 the Euclidean sphere S3. Indeed, he proved that the lower bound is attained on
1. the Clifford torus. Following Urbano’s result, El Soufi [5] proved that the index of
0 a compact non-totally geodesic minimal hypersurface of the sphere not only must
9 be greater than one, but in fact, must be greater than or equal to n+3 which cor-
0
respond to the index of a minimal Clifford torus. Therefore, we have the following
:
v result.
i
X Theorem 2.2 [El Soufi]. Let Σn be a compact orientable minimal hypersurface
r immersed into Sn+1. Then
a
(1) either Ind(Σ)=1 (and Σ is a totally geodesic equator Sn Sn+1);
⊂
(2) or Ind(Σ) n+3.
≥
Ontheotherhand,apartfromtotallygeodesicequatorsofthesphere,theeasiest
minimalhypersurfacesinSn+1 aretheminimalCliffordtori. Forthatreason,ithas
beenconjecturedthatthe minimalCliffordtoriaretheonlyminimalhypersurfaces
of the sphere with index n+3. Thus, we have the following.
Conjecture 1. Let Σn be a compact orientable minimal hypersurface immersed
into Sn+1. Then
Date:January28,2009.
1991 Mathematics Subject Classification. Primary53C42,53C20; Secondary53C65.
Key words and phrases. Constantmeancurvature,Index, Stability.
SupportedbyNSF-OISE:0526008andCNPq: 451778/2008-1.
1
2 E.COLBERG,A.M.DEJESUS,K.KINNEBERG,G.SILVANETO
(1) either Ind(Σ)=1 (and Σ is a totally geodesic equator Sn Sn+1);
⊂
(2) or Ind(Σ) n+3,with equality if and only if Σ is a minimal Clifford torus
≥
Sk( k) Sn−k( n−k).
n × n
q q
This conjecture has not yet been proven,but many mathematicians have solved
some particular cases. For instance, as pointed out Urbano [10] proved that this
conjectureistruewhenn=2. Lateron,Brasil,DelgadoandGuadalupe,[4]showed
that the conjecture is true for every dimension n provided the scalar curvature is
constant. But, in fact, this follows from a early result due to Nomizu and Smyth
[7]. On the other hand, Perdomo [8] has also showed that the conjecture is true
for every dimension n with a hypothesis about the symmetry of the hypersurface.
More exactly, he proved.
Theorem 1.3 [Perdomo]. Let Σn be a compact orientable minimal hypersurface
immersed into Sn+1, invariant under the antipodal map, and not a totally geodesic
equator. Then Ind(Σ) n+3, with equality if and only if Σ is a minimal Clifford
≥
torus.
In his paper, Perdomo observed that the symmetry assumption seems weak be-
cause allof the many knownembedded minimal hypersurfacesin dimension bigger
than 3 of the sphere do have antipodal symmetry.
Besides minimal hypersurfaces of Sn+1, a natural generalization is the case of
hypersurfaces with constant mean curvature (CMC) hypersurfaces. In this case, it
is more natural geometrically to study the weak index of such hypersurfaces.
Barbosa, do Carmo and Eschenburg [3] proved that totally umbilical round
spheres Sn(r) Sn+1 are the only compact weakly stable CMC hypersurfaces
in Sn+1. It has⊂also been proven that the minimal Clifford tori have weak index
equal to n+2 when regarded as CMC hypersurfaces. These results lead us to a
natural conjecture.
Conjecture 2. Let Σn be a compact orientable CMC hypersurface immersed into
Sn+1. Then
(1) either Ind (Σ)=0 (and Σ is a totally umbilical sphere in Sn+1);
T
(2) or Ind (Σ) n+2, with equality if and only if Σ is a CMC Clifford torus
T
≥
Sk(r) Sn−k(√1 r2) with radius k r k+2.
× − n+2 ≤ ≤ n+2
q q
As was first done on the minimal case, recently, Al´ıas, Brasil Jr. and Perdomo
[2] showed that this conjecture is true under the additional hypothesis that the
hypersurface has constant scalar curvature.
Inthispaper,weshallpresentalowerboundfortheindexofaCMChypersurface
under an additional hypothesis on the second fundamental form. More exactly, we
obtain the following result.
Theorem 1. Let x : Σn ֒ Sn+1 be a CMC isometric immersion of a compact
→
oriented manifold Σn. Assume that A2 2nH2 0.
| | − ≥
(1) If H = 1 then Ind (Σn) n+2.
T
± ≥
(2) If l 2dΣ n l2dΣ and H 1, then Ind (Σn) n+2.
Σ|∇v| ≥ Σ v | |≤ T ≥
(3) If l 2dΣ n l2dΣ and H 1, then Ind (Σn) n+2.
RΣ|∇v| ≤ RΣ v | |≥ T ≥
R R
ON THE INDEX OF CONSTANT MEAN CURVATURE HYPERSURFACES 3
2. Background
Letusconsiderϕ:Σn ֒ Sn+1,acompactorientablehypersurfaceimmersedinto
the unit Euclideansphere→Sn+1. We willdenote by A the shape operatorofΣ with
respecttoagloballydefinednormalunitvectorfieldN. Thatis,A: (Σ) (Σ)
X →X
is the endomorphism defined by
(2.1) AX = ˜ N = ¯ N,X (Σ)
X X
−∇ −∇ ∈X
where ˜ and ¯ denote,respectively,theLevi-CivitaconnectiononRn+2 andSn+1.
∇ ∇
The mean curvature of Σ is defined as
1
H = tr(A).
n
If denotes the Levi-Civita connection on Σ, then the Gauss formula for the
∇
immersion ϕ is given by
(2.2) ˜ Y = ¯ Y X,Y ϕ= Y + AX,Y N X,Y ϕ
X X X
∇ ∇ −h i ∇ h i −h i
foreverytangentvectorfieldX,Y (Σ). ThecovariantderivativeofAisdefined
∈X
by
A(X,Y)=( A)X = (AX) A( X),X,Y (Σ),
Y Y Y
∇ ∇ ∇ − ∇ ∈X
and the Codazzi equation is given by
(2.3) A(X,Y)= A(Y,X)
∇ ∇
for any X,Y (Σ).
Every smo∈othXfunction f C∞(Σ) induces a normalvariationϕ :Σ Sn+1 of
t
∈ →
the original immersion ϕ, given by
ϕ =cos(tf(p))ϕ(p)+sin(tf(p))N(p).
t
Since ϕ = ϕ is an immersion and this is an open condition, there exists an ε > 0
0
suchthat everyϕ is also an immersion,for t <ε. Then we can consider the area
t
function :( ε,ε) R given by | |
A − →
(t)=Area(Σ )= dΣ ,
t t
A
ZΣ
where Σ stands for the manifold Σ endowed with the metric induced by ϕ from
t t
the Euclidean metric on Sn+1, and dΣ is the n-dimensional area element of that
t
metric on Σ. The first variation formula for the area ([6], Chapter 1, Theorem 4)
establishes that
d
′
(2.4) (0)= [ (t)] = n fHdΣ.
A dt A t=0 −
ZΣ
The stability operatorof this variationproblem is givenby the second variation
formula for the area ([6], Chapter 1, Theorem 32),
d2
(2.5) ′′(0)= [ (t)] = f(∆f + A2f +nf)dΣ= fJfdΣ.
A dt2 A t=0 − | | −
ZΣ ZΣ
Here, J = ∆+ A2+n is the Jacobi operator, where ∆ stands for the Laplacian
| |
operator of Σ and A2 =tr(A2).
| |
Whenworkingwithconstantmeancurvaturehypersurfaces,itisoftenconvenient
tousethetracelesssecondfundamentalformgivenbyφ=A HI,whereI denotes
−
the identity operator on (Σ).
X
4 E.COLBERG,A.M.DEJESUS,K.KINNEBERG,G.SILVANETO
As a consequence of the first variation formula for the area 2.4, we have that
Σ has constant mean curvature (not necessarily zero) if and only if A′(0) = 0 for
every smooth function f C∞(Σ) satisfying the additional condition fdΣ=0.
The Jacobi operator in∈duces the quadratic form Q : C∞(Σ) R actΣing on the
→ R
space of smooth functions on Σ defined by
Q(f)= fJfdΣ.
−
ZΣ
There are two different notions of stability and index, the strong stability and
strong index, denoted by Ind(Σ), and the weak stability and weak index, denoted
by Ind (Σ). Thus, the strong index is
T
∞
Ind(Σ)=max dimV :V C (Σ), Q(f)<0, f V
{ ⊂ ∀ ∈ }
and Σ is called strongly stable if and only if Ind(Σ) = 0. On the other hand, the
weak index is
∞
Ind (Σ)=max dimV :V C (Σ), Q(f)<0, f V ,
T { ⊂ T ∀ ∈ }
where
∞ ∞
C (Σ)= f C (Σ): fdΣ=0 ,
T { ∈ }
ZΣ
and Σ is called weakly stable if and only if Ind Σ=0.
T
2.1. Preliminary Calculations. Let ϕ : Σn ֒ Sn+1 be a constant mean curva-
→
ture isometric immersion of a compact oriented manifold Σn. For a fixed arbitrary
vectorv Rn+2, wewill considerthe supportfunctions l = ϕ,v andf = N,v
v v
∈ h i h i
definedonΣ. Astandardcomputation,usingequations2.1and2.2,showsthatthe
gradient and the Hessian of the functions l and f are given by
v v
(2.6) l =vT
v
∇
(2.7) 2l (X,Y):= l ,Y = l X,Y +f AX,Y
v X v v v
∇ h∇ ∇ i − h i h i
and
(2.8) f = A(vT)
v
∇ −
(2.9) 2f (X,Y):= f ,Y = A(vT,X),Y +l AX,Y f AX,AY ,
v X v v v
∇ h∇ ∇ i −h∇ i h i− h i
for every tangent vector field X,Y (Σ). Here, vT = v l ϕ f N (Σ)
v v
∈ X − − ∈ X
denotes the tangential component of v along the immersion ϕ.
Equation 2.7 directly yields
∆l =tr( 2l )= nl +nHf .
v v v v
∇ −
Using the Codazzi equation 2.3 in equation 2.9 we also obtain
∆f = n vT, H +nHl A2f .
v v v
− h ∇ i −| |
In particular , if H is constant we have
∆f =nHl A2f .
v v v
−| |
From here, a direct calculation yields
Jl = A2l +nHf
v v v
| |
and
Jf =nHl +nf .
v v v
ON THE INDEX OF CONSTANT MEAN CURVATURE HYPERSURFACES 5
3. Results
Consider the function ψ =l Hf . Since φ2 = A2 nH2, we obtain
v v v
− | | | | −
Jψ = φ2l
v v
| |
and
ψ dΣ=0.
v
ZΣ
Then a straightforwardcomputation yields
(3.1) Q(ψ )= φ2l2dΣ+ φ2Hl f dΣ.
v − | | v | | v v
ZΣ ZΣ
Proposition 1. Let ϕ : Σn ֒ Sn+1 be a constant mean curvature isometric im-
→
mersion of a compact oriented manifold Σn. If e ,...,e stand for the canonical
1 n+2
vector field of Rn+2, then
n+2
Q(ψ )= φ2dΣ.
ei − | |
i=1 ZΣ
X
Proof. First, notice that equation 3.1 yields
n+2 n+2 n+2
Q(ψ )= φ2 l2 dσ+ H φ2 l f dσ.
ei − | | ei | | ei ei
i=1 ZΣ i=1 ZΣ i=1
X X X
Taking into account that
n+2 n+2 n+2 n+2
x,x = x,e e , x,e e = x,e 2 = l2
h i * h ii i h ji j+ h ii ei
i=1 j=1 i=1 i=1
X X X X
and
n+2 n+2 n+2 n+2
x,N = x,e e , N,e e = x,e N,e = l f
h i * h ii i h ji j+ h iih ii ei ei
i=1 j=1 i=1 i=1
X X X X
we arrive at
n+2
Q(ψ )= φ2dσ,
ei − | |
i=1 Zσ
X
which finishes the proof of proposition. (cid:3)
Corollary 1. Let ϕ : Σn ֒ Sn+1 be a constant mean curvature isometric im-
→
mersion of a compact oriented manifold Σn. Then, up to totally umbilical spheres,
Ind (Σ) 1.
T
≥
Proof. If Σn is not totally umbilical, then φ2 >0. Thus,
| |
n+2
Q(ψ )<0
ei
i=1
X
so there is at least one ψ such that Q(ψ )<0. Hence, Ind (Σ) 1. (cid:3)
ei ei T ≥
6 E.COLBERG,A.M.DEJESUS,K.KINNEBERG,G.SILVANETO
3.1. Main Result. Our main result is the following theorem.
Theorem 1. Let ϕ : Σn ֒ Sn+1 be a constant mean curvature isometric immer-
→
sion of a compact oriented manifold Σn. Assume that A2 2nH2 0 and Σn is
| | − ≥
not totally umbilical. Then the following are true.
If H = 1, then Ind (Σn) n+2.
T
• If ±l 2dΣ n l2dΣ f≥or all v Rn+2 and H 1, then Ind (Σn)
• Σ|∇v| ≥ Σ v ∈ | |≤ T ≥
n+2.
If R l 2dΣ nR l2dΣ for all v Rn+2 and H 1, then Ind (Σn)
• Σ|∇v| ≤ Σ v ∈ | |≥ T ≥
n+2.
R R
Our first objective is to show that if W = span ψ n+2, where e n+2 is the
canonicalframeofRn+2,then(uptototallyumbilic{alesip}hi=e1res)dimW{ =i}in=+1 2. We
will then use this fact to prove our main result.
Lemma 1. Let ϕ:Σn ֒ Sn+1 be a constant mean curvature isometric immersion
→
of a compact oriented manifold Σn. Then, up tototally umbilical spheres, dimW =
n+2.
Proof. Letussupposethat ψ ,...,ψ isadependentset,where e ,...,e
isthecanonicalframeofRn{+2.e1Then,tehne+r2e}existnon-nullrealconstan{ts1a ,...,na+2}
1 n+2
such that
n+2
a ψ =0.
i ei
i=1
X
Thus, considering v = n+2a e , we conclude that l =Hf . We then know that
i=1 i i v v
0 = ∆(l Hf ) = nl +nHf +H A2f nH2l = φ2l . We now claim
v v v v v v v
that this y−ields φ2 =−P0. Indeed, if there| e|xists−some p Σ|su|ch that φ2(p) = 0,
| | ∈ | | 6
then there is a neighborhood of p such that φ2(q) = 0 for all q . Thus,
l (q) = 0 for all q . Hence,Uϕ(Σn) = Sn+1 on| |. Sin6ce H is consta∈ntU, we have
v
ϕ(Σn) = Sn, so ϕ(∈ΣU) is totally umbilical. TherefUore, we have φ2 0. But this
| | ≡
also implies that ϕ(Σ) is totally umbilical. Hence, up to totally umbilical spheres,
ψ ,...,ψ is an independent set. (cid:3)
{ e1 en+2}
We now prove the main theorem.
Proof. By Lemma 1, it suffices to show that Q(ψ ) < 0 for all v Rn+2. We
v
∈
observe that l ∆f = A2l f +nHl2, so
v v −| | v v v
H A2l f dΣ= H l ∆f dΣ+nH2 l2dΣ.
| | v v − v v v
ZΣ ZΣ ZΣ
Also, by Green’s formula and the compactness of Σn, l ∆f dΣ = f ∆l dΣ,
Σ v v Σ v v
so
R R
H A2l f dΣ= H f ∆l dΣ+nH2 l2dΣ.
| | v v − v v v
ZΣ ZΣ ZΣ
ON THE INDEX OF CONSTANT MEAN CURVATURE HYPERSURFACES 7
Therefore, we have the following.
Q(ψ ) = φ2l2dΣ+ φ2Hl f dΣ
v − | | v | | v v
ZΣ ZΣ
= φ2l2dΣ+ H A2l f dΣ n H3l f dΣ
− | | v | | v v − v v
ZΣ ZΣ ZΣ
= φ2l2dΣ H f ∆l dΣ+nH2 l2dΣ nH3 l f dΣ
− | | v − v v v − v v
ZΣ ZΣ ZΣ ZΣ
= φ2l2dΣ+nH(1 H2) l f dΣ+nH2 (l2 f2)dΣ
− | | v − v v v− v
ZΣ ZΣ ZΣ
= (A2 2n)l2dΣ n f2dΣ
− | | − v − v
ZΣ ZΣ
n f2dΣ,
≤ − v
ZΣ
where the last two inequalities come from our assumptions that H = 1 and
A2 2nH2 0. If f 0 for all v Rn+2, then we have Q(ψ ) < 0±for all
v v
| | − ≥ 6≡ ∈
ψ W, as desired. Now, if f 0 for some v, then by ([7], Theorem 1) we
v v
∈ ≡
know that Σn is totally geodesic. This concludes the proof of the first part of the
theorem. To prove the second and third parts of the theorem, let us consider the
expansion
Q(ψ )= (φ2 nH2)l2dΣ nH2 f2dΣ+nH(1 H2) l f dΣ.
v − | | − v − v − v v
ZΣ ZΣ ZΣ
By our assumption that A2 2nH2, we know that (φ2 nH2)l2dΣ 0.
| | ≥ − Σ | | − v ≤
Also, if Σn is not totally geodesic, nH2 f2dΣ < 0. We now want to estimate
− Σ v R
the third term in the expansion. To do this, recall that
R
l ∆l = nl2+nHf l
v v − v v v
and
1
∆(l2)=l ∆l + l 2.
2 v v v |∇v|
Then
nH f l dΣ= l 2dΣ+n l2dΣ.
v v − |∇v| v
ZΣ ZΣ ZΣ
Using the hypotheses in the second and third parts, we find that
nH(1 H2) f l dΣ 0.
v v
− ≤
ZΣ
Hence,underourassumptions,Q(ψ )<0forallv,sothesecondandthirdparts
v
are proved. (cid:3)
4. Acknowledgments
The authors thank their adviser, Professor Abdˆenago Barros, for his invaluable
help, without which this paper would not have been possible. They also thank
ProfessorMariaHelenaNoronha,whocoordinatedthe2008InternationalResearch
Experiencefor Students (IRES)duringwhichthe researchforthis paperwasdone.
8 E.COLBERG,A.M.DEJESUS,K.KINNEBERG,G.SILVANETO
The authors also gratefully acknowledge support from the National Science Foun-
dation (NSF-OISE Grant 0526008)and Brazil’s Conselho Nacional de Desenvolvi-
mento Cient´ıfico e Tecnol´ogico (CNPq Grant 451778/2008-1).
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