The $f$-Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons

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by  Hilário Alencar

THE f-STABILITY INDEX OF THE CONSTANT WEIGHTED MEAN
CURVATURE HYPERSURFACES IN GRADIENT RICCI SOLITONS
HILA´RIO ALENCAR AND ADINA ROCHA
Abstract. In this paper, we prove that a noncompact complete hypersurface with finite
weightedvolume,weightedmeancurvaturevectorboundedinnorm,andisometricallyimmersed
7 in a complete weighted manifold is proper. In addition, we obtain an estimate for f-stability
1
indexofaconstantweightedmeancurvaturehypersurfacewithfiniteweightedvolumeandiso-
0
2 metrically immersed in a shrinking gradient Ricci soliton that admits at least one parallel field

n globally defined. For such hypersurface, we still give a necessary condition for equality to be
a
J achieved in the estimate obtained.

2

]
G
D 1. Introduction
.
h
n+1
t Consider a (n + 1)-dimensional Riemannian manifold (M ,g) endowed with a weighted
a
m measure of the form e−fdµ, where f is a smooth function on M and dµ is the volume element
[
induced by the Riemannian metric g. A weighted manifold is a triple

1
v Mn+1 = (Mn+1,g,e−fdµ).
3 f
7
3 A natural extension of the Ricci tensor to this new context is the Bakry-E´mery Ricci tensor, see
0
0 [1], given by
.
1 2
Ric = Ric+∇ f,
0 f
7 2 n+1
where ∇ f is the Hessian of f on M . It is known that a complete weighted manifold
1
:
v satisfying Ricf ≥ kg for some constant k > 0 is not necessarily compact. One of the examples
Xi is the Gaussian shrinking soliton (cid:18)Rn+1,gcan,e−|x4|2dµ(cid:19) with the canonical metric gcan and
r
a 1
Ric = g .
f can
2
The gradient Ricci solitons are natural generalizations of the Einstein metrics and was intro-
n+1
ducedbyHamiltonin[17]. Indeed,acompleteRiemannianmetricg onasmoothmanifoldM
The first author was partially supported by CNPq of Brazil. The second author was partially supported by
CAPES of Brazil.
Date: December 20, 2016.
1991 Mathematics Subject Classification. 58J50; 53C42; 58E30.
Key words and phrases. Proper hypersurface; Weighted volume; Constant weighted mean curvature; Index of
f-stability operator; Parallel field.
1
2 HILA´RIOALENCARANDADINAROCHA
is a shrinking gradient Ricci soliton if there exists a potential function f, and a real constant
k > 0 such that the Ricci tensor Ric of the metric g satisfies the equation
2
(1.1) Ric+∇ f = kg,
2
where ∇ f denotes the Hessian of f. In this context, the gradient Ricci solitons are complete
weighted manifolds with Ric = kg for some constant real k.
f
Observe that when the potential function is a constant, the gradient Ricci solitons are simply
Einstein metrics. It is still important to mention that gradient Ricci solitons plays an important
role in Hamilton’s Ricci flow and correspond to self-similar solutions, and often arise as Type I
singularity models. For more details see [11].
Let x : Mn → Mn+1 be an isometric immersion of a Riemannian orientable manifold Mn
f
into weighted manifold Mn+1. The function f : M → R, restricted to M, induces a weighted
f
measure e−fdσ on M. Thus, we have an induced weighted manifold Mn = (M,(cid:104), (cid:105),e−fdσ).
f
The second fundamental form A of x is defined by
A(X,Y) = (∇ Y)⊥, X,Y ∈ T M, p ∈ M,
X p
where⊥symbolizestheprojectionabovethenormalbundleofM. Theweighted mean curvature
vector of M is defined by
H = H+(∇f)⊥,
f
and its the weighted mean curvature H is given by
f
H = −H η,
f f
where H = trA and η is unit outside normal vector field. The hypersurface M is called f-
minimal when its weighted mean curvature vector H vanishes identically, and when there
f
exists real constant C such that H = −Cη, we say the hypersurface M has constant weighted
f
mean curvature.
The weighted volume of a measurable set Ω ⊂ M is given by
ˆ
Vol (Ω) = e−fdσ.
f
Ω
Let BM be the geodesic ball of M with center in a fixed point o ∈ M and radius r > 0. It
r
is said that the weighted volume of M has polynomial growth if there exists positive numbers α
and C such that
(1.2) Vol (BM ∩M) ≤ Crα
f r
f-STABILITY INDEX OF THE CONSTANT WEIGHTED MEAN CURVATURE HYPERSURFACES 3
for any r ≥ 1. When α = n in (1.2), M is said to have Euclidean volume growth.
We can consider either f-minimal or constant weighted mean curvature hypersurfaces in
gradient Ricci solitons. In particular, a self-shrinker to the mean curvature flow is a f-minimal
hypersurface of the shrinking Gaussian soliton. In [4], Cheng and Zhou showed that for f-
(cid:18) (cid:19)
minimal hypersurfaces in the shrinking Gaussian soliton Rn+1,gcan,e−|x4|2dx , the properness
of its immersion, its polynomial volume growth, and its finite weighted volume are equivalent
to each other. Those equivalences are still being valid for f-minimal hypersurfaces immersed in
1
a complete shrinking gradient Ricci soliton M satisfying Ric = g, where g is Riemannian
f f
2
metric and f is a convex function (see [2]). In this direction, the following result was obtained:
Proposition 1.1. Let Mn be a noncompact complete hypersurface isometrically immersed in a
complete weighted manifold Mm, with weighted mean curvature vector bounded in norm. If Mn
f
has finite weighted volume, then Mn is proper.
Proposition 1.2. Let f ∈ C∞(M) be a convex function, Mm a gradient Ricci soliton with
f
1
Ric = g, and x : Mn → Mm a noncompact complete immersion with weighted mean curvature
f 2 f
vector satisfying
sup(cid:104)H ,∇f(cid:105) < ∞.
f
x∈M
If Mn is proper, then it has finite weighted volume and Euclidean volume growth.
Now it is known that the weighted Laplacian operator ∆ , defined by
f
∆ u := ∆u−(cid:104)∇f,∇u(cid:105),
f
is associated to e−fdσ as well as ∆ is associated to dσ. Moreover, ∆ is a self-adjoint operator
f
on the L2 space of square integrable functions on M with respect to the measure e−fdσ, and
f
therefore, the L2 spectrum of ∆ on M, denoted by σ(−∆ ), is a subset of [0,+∞).
f f f
Next, let F : (−ε,ε) × M → M , F (p) = F(t,p) for all t ∈ (−ε,ε) and p ∈ M be a
f f
variation of the immersion x associated with the normal vector field uη, where u ∈ C∞(M).
c
The corresponding variation of the functional weighted area A (t) = Vol (F (M)) satisfies
f f t
ˆ
(1.3) A(cid:48) (0) = H ue−fdσ,
f f
M
where H is such that H = −H η. The expression (1.3) is known as first variation formula.
f f f
The f-minimal hypersurfaces are critical points of the functional weighted area. Yet, the
hypersurfaces with constant weighted mean curvature can be viewed as critical points of the
functional weighted area restricted to variations which preserve the enclose weighted volume,
4 HILA´RIOALENCARANDADINAROCHA
i.e., for functions u ∈ C∞(M) which satisfy the additional condition
c
ˆ
ue−fdσ = 0.
M
For such critical points, the second variation of the functional weighted area is given by
ˆ
A(cid:48)(cid:48)(0) = − (cid:0)u∆ u+(cid:0)|A|2+Ric (η,η)(cid:1)u2(cid:1)dσ,
f f f
M
where Ric is the Bakry-E´mery Ricci curvature and A is the second fundamental form. For
f
more details, see [10].
Remark 1.1. When f is a constant function, the first and second variation formula were given
by Barbosa and do Carmo [7] and Barbosa, do Carmo and Eschenburg [6].
The operator
L = ∆ +|A|2+Ric (η,η)
f f f
is called the f-stability operator of the immersion x. In the f-minimal case, the f-stability
operator is viewed as acting on F = C∞(M); in the case of the hypersurfaces with constant
c
weighted mean curvature, the f-stability operator is viewed as acting on
ˆ
(cid:26) (cid:27)
F = C∞(M)∩ u ∈ C∞(M); ue−fdσ = 0 .
c c
M
Associated with L is the quadratic form
f
ˆ
I (u,u) = − uL ue−fdσ.
f f
M
ForeachcompactdomainΩ ⊂ M, definetheindex,Ind Ω, ofL inΩasthemaximaldimension
f f
of a subspace of F where I is a negative definite. The index, Ind M, of L in M (or simply,
f f f
the index of M) is then defined by
Ind M = sup Ind Ω,
f f
Ω⊂M
where the supreme is taken over all compact domains Ω ⊂ M. For more details, see [14] and [5].
Let M ⊂ Rn+1 be a proper, non-planar, two-sided hypersurface satisfying Vol (M) < ∞,
f
1
H = (cid:104)x,η(cid:105)+C and Ind (M) ≤ n, where H is the mean curvature, x is the position vector of
f
2
Rn+1, η is the unit normal field of the hypersurface, Ind (M) is the f-stability index and C is
f
a real constant. McGonagle and Ross ([8], Theorem 5.6) showed that exists a natural number i
such that n+1−Ind (M) ≤ i ≤ n e Σ = Σ ×Ri. In addition, they obtained Ind (M) ≥ 2.
f 0 f
f-STABILITY INDEX OF THE CONSTANT WEIGHTED MEAN CURVATURE HYPERSURFACES 5
It is important to mention that the properness hypothesis can be removed of the Theorem
5.6 of [8]. In fact, by Proposition 1.1, the finite weighted volume implies in the properness of its
immersion.
Next, we obtain an estimate for f-stability index of a constant weighted mean curvature
hypersurface with finite weighted volume and isometrically immersed in a gradient Ricci soliton
that admit at least one parallel field globally defined. If fact,
Theorem 1.1. Let M be a shrinking gradient Ricci soliton with Ric = kg. Let Mn be a
f f
constant weighted mean curvature hypersurface with finite weighted volume and isometrically
n+1
immersed in M . Denote by P the set of parallel fields globally defined on M and η the
f M f
f
unit normal field to M.
(i) If the unit function 1 (cid:54)∈ {(cid:104)X,η(cid:105) : X ∈ P },
M
f
(1.4) Ind (M) ≥ dimP −dim{X ∈ P : (cid:104)X,η(cid:105) ≡ 0}.
f M M
f f
(ii) If the unit function 1 ∈ {(cid:104)X,η(cid:105) : X ∈ P }, M is totally geodesic.
M
f
As a consequence of Theorem 1.1, we have
Corollary 1.1. Let Mn be a constant weighted mean curvature hypersurface with finite weighted
volume and isometrically immersed in a shrinking gradient Ricci soliton M . Denote by P
f M
f
n+1
the set of parallel fields globally defined on M and let η the unit normal field to M. If the
f
unit function 1 (cid:54)∈ {(cid:104)X,η(cid:105) : X ∈ P } and there exist a parallel field X such that (cid:104)X ,η(cid:105) (cid:54)≡ 0,
M 0 0
f
then
Ind (M) ≥ 1.
f
Moreover,
dim{X ∈ P : (cid:104)X,η(cid:105) ≡ 0} = dimP −1
M M
f f
whenever Ind (M) = 1.
f
A necessary condition for equality to be achieved in the estimate (1.4) of Theorem 1.1, is
given by
Theorem 1.2. Let M be a shrinking gradient Ricci soliton with Ric = kg. Let Mn be a
f f
constant weighted mean curvature hypersurface isometrically immersed in M . Denote by P
f M
f
n+1
the set of parallel fields globally defined on M and η the unit normal field to M. Suppose
f
that Vol (M) < ∞, Ind (M) < ∞, dimP > 0, and
f f M
f
Ind (M) = dimP −dim{X ∈ P : (cid:104)X,η(cid:105) ≡ 0}.
f M M
f f
6 HILA´RIOALENCARANDADINAROCHA
(i) If Ind (M) = dimP , M is totally geodesic and the bottom µ (M) of the spectrum of
f M 1
f
f-stability operator satisfies µ (M) = −k.
1
(ii) If Ind (M) (cid:54)= dimP , either M is diffeomorphic to the product of a Euclidian space
f M
f
with some other manifold or there is a circle action on M whose orbits are not real
homologous to zero.
2. Properness and Finite Weighted Volume of a Constant Weighted Mean
Curvature Hypersurface
We will begin by proving Proposition 1.1 which states that a noncompact complete hyper-
surface with finite weighted volume, weighted mean curvature vector bounded in norm, and
isometrically immersed in a complete weighted manifold is proper. Furthermore, long after we
will prove Proposition 1.2, it give some conditions to that the a proper noncompact complete
hypersurface have finite weighted volume.
Proof of Proposition 1.1. We supposed that M is not proper. Thus, there exists a positive real
M M
number R such that B (o)∩M is no compact in M, where B (o) denotes the closure of the
R R
BM(o). Then, for any a > 0 sufficiently small with a < 2R, there exists a sequence {p } of the
R k
points in BM(o)∩M with dist (p ,p ) ≥ a > 0 for any different k and j.
R M k j
Since BM (p )∩BM (p ) = ∅ for any k (cid:54)= j, we obtain BM (p ) ⊂ BM(o), where BM (p )
a/2 k a/2 j a/2 j 2R a/2 k
and BM (p ) denote the intrinsic balls of M of radius a/2, center in p and p , respectively.
a/2 j k j
Let{e ,e ,...,e }beaorthonormalbasisofT M. Ifx ∈ BM (p ),thenthefunctionextrinsic
1 2 n x a/2 j
distance to p , denoted by r (x) = dist (x,p ), satisfies
j j M j
∇2r (e ,e ) = (cid:104)∇ ∇r ,e (cid:105) = (cid:104)∇ ∇r ,e (cid:105)+(cid:104)∇ (∇r )⊥,e (cid:105)
j i i ei j i ei j i ei j i
= (cid:104)∇ ∇r ,e (cid:105)+(cid:104)(∇ ∇r )⊥,e (cid:105)−(cid:104)A(e ,e ),∇r (cid:105)
ei j i ei j i i i j
= ∇2r (e ,e )−(cid:104)H,∇r (cid:105).
j i i j
Observe that M has bounded locally geometry, this is, there exists positive real numbers k and
i so that the sectional curvature of M is bounded above by k and the injectivity radius of M
0
is bounded below by i in a neighborhood of a point o ∈ M. By choosing R > 0 such that
0
√
2R < min{i ,1/ k}, it follows from Hessian comparison of the distance (see Lemma 7.1, [3]),
0
that
√ 1
∇2r (e ,e ) ≥ − k+ |e −(cid:104)e ,∇r (cid:105)∇r |2
j i i i i j j
r
j
f-STABILITY INDEX OF THE CONSTANT WEIGHTED MEAN CURVATURE HYPERSURFACES 7
M M
in B (o). Hence, in B (o)∩M,
2R 2R
n n
∆r = (cid:88)∇2r (e ,e ) = (cid:88)∇2r (e ,e )+(cid:104)H,∇r (cid:105)
j j i i j i i j
i=1 i=1
(cid:88)n (cid:18) √ 1 (cid:19)
≥ − k+ |e −(cid:104)e ,∇r (cid:105)∇r |2 +(cid:104)H,∇r (cid:105)+(cid:104)(∇f)⊥,∇r (cid:105)−(cid:104)(∇f)⊥,∇r (cid:105)
i i j j j j j
r
j
i=1
√ n |∇r |2
= −n k+ − j +(cid:104)H ,∇r (cid:105)−(cid:104)(∇f)⊥,∇r (cid:105)
f j j
r r
j j
√ n |∇r |2
≥ −n k+ − j −|H |−|(∇f)⊥|.
f
r r
j j
Byhypothesis,thenormofH isboundedaboveandsup |∇f(p)| < ∞foreachR > 0.
f M
p∈B (o)
2R
Thus,
√ n |∇r |2
∆r ≥ −n k+ − j − sup |H (p)|− sup |(∇f(p))⊥|
j f
r r
j j p∈BM(o)∩M p∈BM(o)∩M
2R 2R
n |∇r |2
j
≥ − −C,
r r
j j
where
√
C = n k+ sup |H (p)|+ sup |∇f(p)|.
f
p∈B2MR(o)∩M p∈BM2R(o)
Therefore, in BM(o)∩M,
2R
(cid:18)n |∇r |2 (cid:19)
∆r2 = 2r ∆r +2|∇r |2 ≥ 2r − j −C +2|∇r |2 = 2n−2Cr .
j j j j j r r j j
j j
By choosing a < min{2n/C,2R}, we have for 0 < ζ ≤ a/2,
ˆ ˆ ˆ
(2n−2Cr )dσ ≤ ∆r2dσ = (cid:104)∇r2,ν(cid:105)dσ
j j j
BζM(pj) ˆBζM(pj) ∂BζM(pjˆ)
(2.1) ≤ 2r |∇r ||ν|dA ≤ 2r dA
j j j
∂BζM(pj) ∂BζM(pj)
≤ 2ζA (ζ),
j
8 HILA´RIOALENCARANDADINAROCHA
where ν denotes the unit normal vector field pointing out of ∂BM(p ) and A (ζ) denotes the
ζ j j
area of ∂BM(p ). By using co-area formula in (2.1), we have
ζ j
ˆ ˆ ˆ
ζ
(n−Cr )dσ = (n−Cr )|∇r |−1dA dt
j j j t
BζM(pj) ˆ0 ˆ∂BtM(pj)
ζ
≥ (n−Cr )dA dt
j t
ˆ0 ∂BtM(pj)ˆ
ζ
≥ (n−Ct) dA dt
t
0 ∂BtM(pj)
≥ (n−Cζ)V (ζ),
j
where V (ζ) denotes the volume of BM(p ). Therefore,
j ζ j
(2.2) (n−Cζ)V (ζ) ≤ ζA (ζ).
j j
Since
ˆ ˆ ˆ ˆ
d d ζ
V(cid:48)(ζ) = dσ = |∇r |−1dA dt = |∇r |−1dA ≥ A (ζ),
j dσ dσ j t j j
BζM(pj) 0 ∂BtM(pj) ∂BζM(pj)
then
(n−Cζ)V (ζ) ≤ ζV(cid:48)(ζ).
j j
Thus,
d V(cid:48)(ζ) n
j
(2.3) logV (ζ) = ≥ −C.
j
dσ V (ζ) ζ
j
By integrating (2.3) from ε > 0 to ζ, we obtain
logV (ζ)−logV (ε) ≥ nlogζ −nlogε−C(ζ −ε),
j j
that is
V (ζ) ζn
j ≥ e−C(ζ−ε).
V (ε) εn
j
Now observing that
V (ε)
j
lim = ω ,
ε→0+ εn n
we obtain
V (ζ) ≥ ω ζne−Cζ
j n
f-STABILITY INDEX OF THE CONSTANT WEIGHTED MEAN CURVATURE HYPERSURFACES 9
for 0 < ζ ≤ a/2. Thus, we conclude that
ˆ ˆ
∞
(cid:88)
Vol (M) = e−f dσ ≥ e−f dσ
f
M j=1 BaM/2(pj)
(cid:32) (cid:33) ∞
(cid:88)
≥ inf e−f V (a/2) = +∞.
j
M
B2R(o) j=1
This contradicts the assumption of the finite weighted volume of M. Therefore, Mn is a proper
hypersurface of Mm. (cid:3)
f
Definition 2.1. The function f ∈ C∞(M) is said convex if the Hessian of f is non-negative,
2
this is ∇ f ≥ 0.
1
Remark 2.1. Let M be a complete gradient Ricci soliton satisfying Ric = g. In this case,
f f
2
Cao and Zhou [16] showed that by translating f
(2.4) R+|∇f|2−f = 0 e R ≥ 0.
Thus, it follows from the equations in (2.4) that
(2.5) |∇f|2 ≤ f.
In addition, there exists constants c ,c ∈ R such that
1 2
1 1
(2.6) (r(x)−c )2 ≤ f(x) ≤ (r(x)+c )2,
1 2
4 4
where r(x) = dist (x,o) is the distance of x ∈ M to a fixed point o ∈ M. The constant c
M 2
depends only on the dimension of the manifold and c depends on the geometry of g on unit ball
1
center in o (see [16], Lemma 2.1, Lemma 2.2 and Theorem 1.1). In [9], Munteanu and Wang
1
showed that the inequalities in (2.6) are true only assuming that Ric ≥ g and |∇f|2 ≤ f.
f
2
1
Proof of Proposition 1.2. By hypothesis, Ric = g. Thus, it follows from Remark 2.1 that
f
2
m
R+|∇f|2−f = 0, R+∆f = , and R ≥ 0,
2
and we have that
m
∆f −|∇f|2+f = and |∇f|2 ≤ f.
2
10 HILA´RIOALENCARANDADINAROCHA
By being f a convex function, we obtain
∆ f +f = ∆f −|∇f|2+f = ∇2f(e ,e )+(cid:104)H,∇f⊥(cid:105)−|∇f|2+f
f i i
m
= ∆f − (cid:88) ∇2f(η ,η )+(cid:104)H ,∇f⊥(cid:105)−|∇f⊥|2−|∇f|2+f
i i f
i=n+1
m
= ∆f − (cid:88) ∇2f(η ,η )+(cid:104)H ,∇f⊥(cid:105)−|∇f|2+f
i i f
i=n+1
m
≤ +C,
2
where C = sup (cid:104)H ,∇f⊥(cid:105) < ∞.
x∈M f
Observe that
1 1
(2.7) (r(x)−c)2 ≤ f(x) ≤ (r(x)+c)2,
4 4
where c is a constant (see Remark 2.1, inequalities in (2.6)). Hence, we can conclude that f is
proper on M. Since, by hypothesis, x : M → M is a proper immersion, then f| : M → R is
f M
a proper function.
Therefore,itfollowsfromTheorem1.1de[4]thatM hasfiniteweightedvolumeandEuclidean
volume growth of the sub-level set of the potential function f. (cid:3)
3. The f-stability Index of the Constant Weighted Mean Curvature
Hypersurfaces
In this section, we will prove Theorem 1.1, Corollary 1.1, and Theorem 1.2. Which are results
about the f-stability index of a constant weighted mean curvature hypersurface with finite
weighted volume and isometrically immersed in a gradient Ricci soliton that admits at least one
parallel field globally defined. For this, we are going to give some definitions and known results.
Definition 3.1. We say that a vector field X ∈ TM is parallel if
∇ X = 0
Y
for all vector fields Y ∈ TM.
n+1
Proposition 3.1. Let M be a gradient Ricci soliton satisfying Ric = kg and X a parallel
f f
vector field on Mn+1. If Mn is a constant weighted mean curvature hypersurface immersed
f
n+1
isometrically immersed in M , then
f
L (cid:104)X,η(cid:105) = k(cid:104)X,η(cid:105)
f