LOCAL RIGIDITY, BIFURCATION, AND STABILITY OF Hf -HYPERSURFACES IN WEIGHTED KILLING WARPED PRODUCTS

In a weighted Killing warped productMn f ×ρR with warping metric 〈 , 〉M+ ρ2 dt, where the warping function ρ is a real positive function defined on Mn and the weighted function f does not depend on the parameter t ∈ R, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants, or the local rigidity for a family of open sets {Ωγ}γ∈I whose boundaries ∂Ωγ are hypersurfaces with constant weighted mean curvature. For this, we analyze the number of negative eigenvalues of a certain Schrödinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface x : Σn ↪→Mn f ×ρ R with constant weighted mean curvature in terms of the first eigenvalue of the f -Laplacian of Σn. 2010 Mathematics Subject Classification: Primary: 58J55, 35B32, 53C42; Secondary: 35P15.


Introduction and statements of the results
According to Barbosa and do Carmo in [5], and Barbosa, do Carmo, and Eschenburg in [6], any closed hypersurface Σ n with constant mean curvature (CMC) in a Riemannian manifold M n+1 (n ≥ 2) is a critical point of the variational problem of minimizing the area functional for volume-preserving variations. Moreover, when M n+1 has constant sectional curvature c, they also established that geodesic spheres are the only stable critical points for this variational problem. As observed in [2,9,10], the set of trial maps for the variational problem should be a collection of embeddings of CMC hypersurfaces Σ n into M n+1 . In order to detect solutions that are not isometrically congruent, one should take into consideration the action of the diffeomorphism group of Σ n , acting by right composition in the space of embeddings, and the action of the isometry group of M n+1 , acting by left composition on the space of embeddings. Note that the area and the volume functionals are invariant by the action of these two groups. The action of the diffeomorphism group of Σ n on any set of embeddings of CMC hypersurfaces Σ n into M n+1 is free, which suggests that one should consider a quotient of the space of embeddings by this action. This means that two embeddings of CMC hypersurfaces x 1 : Σ n → M n+1 and x 2 : Σ n → M n+1 will be considered equivalent if there exists a diffeomorphism φ : Σ n → Σ n such that x 2 = x 1 • φ. As to the left action of the isometry group of M n+1 , this is not free; nevertheless, the group is compact, and one can study a bifurcation problem for its critical orbits. Thus, the variational problem described above provides us with a framework where we can study the equivariant bifurcation (cf. [2,10,9,28]) in a set of equivalence classes of embeddings of CMC hypersurfaces Σ n into M n+1 .
In this context, Alías and Piccione in [2] studied the bifurcation of CMC Clifford torus of the form x n,j r : S j (r) × S n−j ( √ 1 − r 2 ) → S n+1 in unit Euclidean sphere S n+1 , where j ∈ {1, . . . , n} and r ∈ (0, 1). More precisely, they showed that the existence of two infinite sequences x n,j ri : S j (r i ) × S n−j ( 1 − r 2 i ) → S n+1 and x n,j s l : S j (s l ) × S n−j ( 1 − s 2 l ) → S n+1 that are not isometrically congruent to the CMC Clifford torus, and accumulating at some CMC Clifford torus, where {r i } i≥3 , {s l } l≥3 ⊂ (0, 1), are sequences of real numbers such that lim i→∞ r i = 1 and lim l→∞ s l = 0. Furthermore, they also showed that for all other values of r ∈ (0, 1) the family of CMC Clifford torus x n,j r : S j (r)×S n−j ( √ 1 − r 2 ) → S n+1 is locally rigid, in the sense that any CMC embedding of S j (r) × S n−j ( √ 1 − r 2 ) into S n+1 which is sufficiently close to x n,j r must be isometrically congruent to an embedding of the CMC Clifford family. Later, de Lima, de Lira, and Piccione ( [16]) adapted the methods of [2] to obtain bifurcation and local rigidity results for a family of CMC Clifford torus in 3-dimensional Berger spheres S 3 τ , with τ > 0. More recently, Koiso, Palmer, and Piccione ( [23]) proved bifurcation results for (compact portions of) nodoids in the 3-dimensional Euclidean space R 3 , whose boundary consists of two fixed coaxial circles of the same radius lying in parallel planes. Moreover, the same authors provide in [24] criteria for the existence of bifurcation branches of fixed boundary CMC surfaces in R 3 and they discuss stability/instability issues for the surfaces in bifurcating branches.
Meanwhile, García-Martínez and Herrera in [20] deduced some bifurcation and local rigidity results for a certain family of CMC hypersurfaces in a class of Riemannnian warped products of the form (I × ρ M n , dt 2 + ρ 2 , M ), namely, in product manifolds I × M n endowed with the warping metric dt 2 + ρ 2 , M , where I ⊂ R is an open interval, ρ is a real positive function defined on I, called warping function, and M n is a closed Riemannian manifold with Riemannian metric , M , called Riemannian fiber. Such results are obtained considering some appropriate hypotheses that depend of the behavior of the eigenvalues of Laplacian operator on M n .
On the other hand, on a complete Riemannian manifold M n+1 , let us remember that the classical Laplace operator ∆ on M n+1 can be defined as the differential operator associated to the standard Dirichlet form where | | is the norm induced by the Riemannian metric , of M n+1 , dV is the volume element on M n+1 , L 2 (dV ) denotes the set of measurable functions u on M n+1 such that the Lebesgue integral (with respect to dV ) of |u| 2 exists and is finite, and C ∞ c (M ) is the set of all smooth functions defined in M n+1 with compact support. Now, let f ∈ C ∞ (M ), that will be referred as a weight function. If we replace the measure dV with the weighted measure dσ = e −f dV in the definition of Q, we obtain a new quadratic form Q f , and we will denote by ∆ f the elliptic operator on C ∞ c (M ) ⊂ L 2 (dσ) induced by Q f . In this sense, ∆ f arises as a natural generalization of the Laplacian. It is clearly symmetric, positive, and extends to a positive operator on L 2 (dσ). By Stokes' theorem, . The triple (M n+1 , , , dσ) and the differential operator ∆ f defined above and acting on C ∞ (M ) will be called, respectively, the weighted manifold associated with M n+1 and f , which we simply denote by M n+1 f , and the f -Laplacian. In this setting, we recall that a notion of curvature for weighted manifolds goes back to Lichnerowicz [25,26] and it was later developed by Bakry andÉmery in their seminal work [3], where they introduced the following modified Ricci curvature Ric f = Ric + Hess f , where Ric and Hess are the standard Ricci tensor and the Hessian on M n+1 f , respectively. As is common in the current literature, we will refer to this tensor as being the Bakry-Émery-Ricci tensor of M n+1 f . We note that the interplay between the geometry of M n+1 and the behavior of the weighted function f is mostly taken into account by means of its Bakry-Émery-Ricci tensor Ric f (cf. [29]).
On the other hand, it is well known that Killing vector fields are important objects which have been widely used in order to understand the geometry of submanifolds and, more particularly, of hypersurfaces immersed in Riemannian spaces. Into this branch, Alías, Dajczer, and Ripoll ( [1]) extended the classical Bernstein's theorem [8] to the context of complete minimal surfaces in Riemannian spaces of nonnegative Ricci curvature carrying a Killing vector field. This was done under the assumption that the sign of the angle function between a global Gauss mapping and the Killing vector field remains unchanged along the surface. Afterwards, Dajczer, Hinojosa, and de Lira ( [15]) defined a notion of Killing graph in a class of Riemannian manifolds endowed with a Killing vector field and solved the corresponding Dirichlet problem for prescribed mean curvature under hypothesis involving domain data and the Ricci curvature of the ambient space. Later on, Dajczer and de Lira ( [13]) showed that an entire Killing graph of constant mean curvature contained in a slab must be a totally geodesic slice, under certain restrictions on the curvature of the ambient space. More recently, in [14] these same authors revisited this thematic treating the case when the entire Killing graph of constant mean curvature contained lies inside a possible unbounded region.
Also recently, the second author, jointly with Cunha, de Lima, Lima, Jr., and Medeiros ( [12]), applied suitable maximum principles in order to obtain Bernstein type properties concerning CMC hypersurfaces Σ n immersed in a Killing warped product (M n × ρ R, , M + ρ 2 dt), namely, in product manifolds M n × R endowed with the warping metric , M + ρ 2 dt, where M n is a Riemannian manifold with Riemannian tensor , M , called Riemannian base, and ρ is a real positive function defined on M n , called warping function. To obtain these results, they assumed that M n satisfies certain constraints and that ρ is concave on M n . Afterwards, in [17] the second author, jointly with Lima, Jr., Medeiros, and Santos, obtained Liouville type results concerning hypersurfaces Σ n immersed in a weighted Killing warped product M n f × ρ R, where the weighted function f does not depend on the parameter t ∈ R. For this, they assumed suitable boundedness on the Bakry-Émery-Ricci tensor of the base M n . Furthermore, they also obtained rigidity results via constraints on the height function of the hypersurface.
Proceeding with the picture described above, our purpose in this paper is to study the notions of local rigidity, bifurcation instants, and stability for a family of open sets {Ω γ } γ of a weighted Killing warped product M n f × ρ R whose boundaries ∂Ω γ are closed hypersurfaces with constant weighted mean curvature H f (γ) (in abbreviation, we say that ∂Ω γ is a closed H f (γ)-hypersurface), where γ varies on a prescribed interval I ⊂ R.
For this, in Section 2 we record some main facts about the hypersurfaces immersed in M n f × ρ R. Next, in Subsection 3.1, for each Ω γ , we establish the variation X : ) of ∂Ω γ and we consider the variational problems: (VP-1): Minimizing the weighted area functional A f (see (3.5)) for all variations of ∂Ω γ that preserve the weighted volume of Ω γ .
(VP-2): Minimizing the weighted area functional A f (see (3.5)) for all variations of ∂Ω γ , not necessarily weighted volume-preserving variations of Ω γ . By an analysis of the first variation of the associated weighted Jacobi functional F , where V f is the weighted volume functional (see (3.4)), we obtain in Proposition 1 that the critical points of (VP-1) and (VP-2) are the open sets Ω γ whose boundary ∂Ω γ is a closed H f (γ)-hypersurface with constant weighted mean curvature H f (γ) = λ(γ)/n. For these critical points, in Proposition 2 we obtain the formula of the second variation of F λ(γ) f . Concerning the variational problem (VP-2), in Subsection 3.2 we use the equivariant bifurcation theory (cf. [2,10,9,28]) to establish our notions of bifurcation instants and local rigidity in terms of the Morse index of the weighted Jacobi operator J f ;γ (see (3.21)). Then, in Section 4 we get some results of local rigidity and bifurcation instants in M n f × ρ R via the analysis the number of negative eigenvalues of J f ;γ . Initially, we establish the following result of local rigidity.
In particular, such a family is locally rigid if one of the following conditions holds: In Theorem 1, Y is the Killing vector field defined on the weighted Killing warped product M n f × ρ R, ρ = |Y | > 0 is the warping function, N γ is the unit normal vector field on ∂Ω γ , ∆ f represents the f -Laplacian on M n f , Ric f and Hess are the Bakry-Émery-Ricci tensor and the Hessian operator on M n f , |A| 2 stands for the square of the norm of the shape operator A of ∂Ω γ with respect to the orientation given by N γ , and N * γ is the orthogonal projection of N onto the tangent bundle of M n . These notations will also be used in the statements of the next theorems.
In turn, the bifurcation instants of the family {Ω γ } γ∈I are established in the following result.
Theorem 2. Let {Ω γ } γ be a family of open subsets of the weighted Killing warped product M n f × ρ R whose boundaries ∂Ω γ are closed H f (γ)hypersurfaces. Suppose that, for all γ ∈ I, the function is constant on ∂Ω γ . If there are two values γ 1 and γ 2 , with γ 1 < γ 2 , such that the eigenvalues µ j f (γ 1 ) and µ j f (γ 2 ) of the weighted Jacobi operators J f ;γ1 and J f ;γ2 (respectively) satisfy Furthermore, in Section 4, when M n is closed Riemannian manifold, we give sufficient conditions for both the existence and nonexistence of bifurcation instants of a certain family {Ω γ } γ of open subsets of the weighted Killing warped product M n f × ρ R (see (4.2)) whose boundaries ∂Ω γ are f -minimal hypersurfaces; namely, each ∂Ω γ is a hypersurface with f -mean curvature equal to zero (cf. Corollaries 1 and 2).
Finally, in Section 5 we study the notion of stability for a critical point of the variational problem (VP-1). More precisely, we established a notion of f -stability for a closed H f -hypersurface Σ n immersed in M n f × ρ R and, with the help of the f -Laplacian ∆ f of Σ n of a certain angle function Θ given in Proposition 3, we obtain the following characterization for the f -stability:

Hypersurfaces in weighted Killing warped products
Unless stated otherwise, all manifold considered in this work will be connected, while closed means compact without boundary. Throughout this paper, we will consider an (n + 1)-dimensional Riemannian manifold M n+1 (n ≥ 2) endowed with a Killing vector field Y . Suppose that the distribution of all vector fields of M n+1 that are orthogonal to Y is of constant rank and integrable. Given an integral leaf M n of that distribution, let Ψ : I × M n → M n+1 be the flow generated by Y with initial values in M n , where I is a maximal interval of definition. Without loss of generality, in what follows we will consider I = R.
In this setting, our space M n+1 can be regarded as the Killing warped product M n × ρ R, that is, the product manifold M n × R endowed with the warping metric , where π M and π R denote the canonical projections from M n × R onto each factor, , M is the induced Riemannian metric on the base M n , dt 2 denotes the usual Riemannian metric in R, and ρ = |Y | > 0 is the warping function. By C ∞ (M n × ρ R) we mean the ring of real functions of class C ∞ on M n × ρ R, and by X(M n × ρ R) the C ∞ (M n × ρ R)-module of vector fields of class C ∞ on M n × ρ R. Let ∇ and ∇ be the Levi-Civita connections of M n × ρ R and M n , respectively. Now, let (M n × ρ R) f be a weighted Killing warped product, namely, a Killing warped product M n × ρ R endowed with a weighted volume form dσ = e −f dv, where f ∈ C ∞ (M n × ρ R) is a real-valued function, called weighted function (or density function), and dv is the volume element induced by the warping metric , defined in (2.1). For (M n × ρ R) f , the Bakry-Émery-Ricci tensor Ric f is defined by where Ric and Hess are the Ricci tensor and the Hessian operator in M n × ρ R, respectively. Throughout this work, we will deal with hypersurfaces x : Σ n → (M n × ρ R) f immersed in a weighted Killing warped product (M n × ρ R) f and which are two-sided. This condition means that there is a globally defined unit normal vector field N . We let ∇ denote the Levi-Civita connection of Σ n .
In this setting, let A denote the shape operator of Σ n with respect to N , so that at each p ∈ Σ n , A restricts to a self-adjoint linear map According to Gromov [21], the weighted mean curvature H f , or simply the f -mean curvature, of x : Σ n → (M n × ρ R) f is given by where H denotes the standard mean curvature of x : Σ n → (M n × ρ R) f with respect to its orientation N . When required, if a hypersurface x : Σ n → (M n × ρ R) f has constant f -mean curvature H f , then for short we will say that x : Σ n → (M n × ρ R) f is an H f -hypersurface. Moreover, we recall that x : Σ n → (M n × ρ R) f is called f -minimal when its f -mean curvature vanishes identically. The f -divergence on Σ n is defined by where div(·) denotes the standard divergence on Σ n . We define the drift Laplacian of Σ n by where ∆ is the standard Laplacian on Σ n . We will also refer to such an operator as the f -Laplacian of Σ n .
Remark 1. We observe that the Killing vector field Y determines in M n × ρ R a codimension one foliation by totally geodesic slices M n × {t}, t ∈ R, with respect to orientation determined by Y . Moreover, assuming that the weighted function f ∈ C ∞ (M n × ρ R) is invariant along the flow determined by Y , that is, ∇f, Y = 0, from (2.3) we get that each slice M n × {t} is f -minimal.
Remark 2. We observe that the following result is a consequence of a Cheeger-Gromoll type splitting theorem due to G. Wei and W. Wylie (cf. Theorem 6.1 of [29]; see also Theorem 1.1 of [19]): Let M Motivated by Remarks 1 and 2, in this work we will consider Killing warped products M n × ρ R endowed with a weighted function f does not depend on the parameter t ∈ R, that is, ∇f, Y = 0. For the sake of simplicity, we will denote such an ambient space by

The variational problem and the notion of bifurcation instants
Let M be the space of open subsets Ω of M n f × ρ R with compact closure Ω and whose smooth compact boundary ∂Ω is a closed, connected, and orientable hypersurface. For any Ω ∈ M, Vol f (Ω) and Area f (∂Ω) will denote the f -volume and f -area of Ω and ∂Ω, respectively.

Description of the variational problem.
If Ω ∈ M, the globally unit normal vector field defined on ∂Ω will be denoted by N . For Ω ∈ M, we define a variation of ∂Ω as being the smooth mapping satisfying the following two conditions: (1) for all s ∈ (− , ), the map is an immersion; (2) X(0, p) = ι(p) for all p ∈ ∂Ω, where ι : ∂Ω → Ω is the inclusion map. In this context, given Ω ∈ M and a variation X : (− , ) × ∂Ω → M n f × ρ R of ∂Ω we adopt the notation ∂Ω s = X s (∂Ω). For values of s small enough, ∂Ω s is also a connected and oriented n-dimensional smooth submanifold. Moreover, it bounds an open subset Ω s whose closure is also compact. Thus, the variation X : (− , ) × ∂Ω → M n f × ρ R described above induces a variation of the open subset Ω denoted by Ω s , which is also an element of M.
In all that follows, we let d(∂Ω s ) denote the volume element of the metric induced on ∂Ω s by X s and N s , the unit normal vector field along X s . Moreover, we also consider in ∂Ω s the weighted volume form given by dσ s = e −f d(∂Ω s ). When s = 0 all these objects coincide with ones defined in ∂Ω, respectively.
The variational field associated to the variation X : where (·) stands for tangential components.
The weighted volume functional associated to the variation X: and we say that X : The following result is well known and, in the context of weighted manifolds, can be found in [11]. Remark 3. We observe that is not difficult to verify that Lemma 2.2 of [6] still remains valid for the context of weighted Riemannian manifolds, that is, if u ∈ C ∞ (∂Ω) is such that ∂Ω u dσ = 0, then there exists a weighted volume-preserving variation X : (− , ) × ∂Ω → M n f × ρ R of ∂Ω whose variational field is ∂X ∂s s=0 = uN . The weighted area functional associated to the variation X is given by Following the same steps of the proof of Lemma 3.2 of [11], it is not difficult to see that we get the following where u s is the function given in (3.3) and (H f ) s = H f (s, ·) denotes the f -mean curvature of ∂Ω s with respect to the metric induced by the immersion X s defined in (3.2).
In order to characterize open subsets Ω of M n f × ρ R whose boundaries are closed hypersurfaces with constant f -mean curvature (possibly equal to zero), we consider the variational problem (VP-1) described in Section 1. The Lagrange multiplier method leads us then to the associated weighted Jacobi functional where λ is a constant to be determined (eventually λ can be zero, and in this case, for Ω ∈ M, our variational problem reduces to minimizing the functional A f for all variations of ∂Ω).
As an immediate consequence of Lemmas 1 and 2 we get that the first variation of F λ f takes the following form H = H f , and this notation will be used in what follows without further comments. Therefore, if we choose λ = nH, from (3.7) we arrive at In particular, Now, from (3.11) and following the same ideas of Proposition 2.7 of [5] we can establish the following result. are open subsets Ω of M n f × ρ R whose boundary ∂Ω is a closed H f -hypersurface with constant second mean curvature H f equal to (3.12) H with λ ∈ R. On the other hand, if we change (VP-1) to (VP-2) (see Section 1), from Proposition 1 we obtain that the respective critical points of (VP-2) coincide with the same critical points of the initial variational problem (VP-1).
Remark 4. If λ = 0, we observe that the two variational problems (VP-2) and (VP-1) coincide, in which case the respective critical points are open subsets Ω of M n f × ρ R whose boundary ∂Ω are closed f -minimal hypersurfaces. Furthermore, from (3.6) we can observe that F 0 f coincides with the weighted area functional A f and, for each Ω ∈ M, this whole situation comes down to the variational problem of minimizing A f for all variations of ∂Ω (not necessarily for those that preserve the weighted volume of Ω).
Remark 5. As observed in [20], our approach is valid for the following more general configuration. Assume that M is the space of open subsets Ω ⊂ M n f × ρ R whose boundary ∂Ω is the union of two disjoint sets ∂Ω = Σ n 1 ∪ Σ n 2 . We will assume that one of them, say Σ n 1 , is a fixed set so that the variations considered of ∂Ω only affect Σ n 2 . Under this assumption, the critical points of (VP-1) or (VP-2) will be open subsets Ω such that their boundaries are the union of a (fixed) set Σ n 1 and a closed H f -hypersurface Σ n 2 with constant f -mean curvature H f given by (3.12).
For such a critical point (for either of the two variational problems described above), the formula for the second variation of F λ f is given in the following result.

Proposition 2.
Let Ω ∈ M be an open subset of M n f × ρ R whose boundary ∂Ω is a compact H f -hypersurface with constant f -mean curvature H f given by (3.12). Then the second variation d 2 ds 2 F λ f (0) of the weighted Jacobi functional F λ f is given by for any u ∈ C ∞ (∂Ω), where J f : C ∞ (∂Ω) → C ∞ (∂Ω) is the weighted Jacobi operator given by Here, Y is the Killing vector field on M n f × ρ R, ρ = |Y | > 0, N is the unit normal vector field on ∂Ω, ∆ f and ∆ f represent the f -Laplacians on ∂Ω and M n f , respectively, Ric f and Hess are the Bakry-Émery-Ricci tensor and the Hessian operator on M n f , |A| 2 represents the square of the norm of the shape operator A of ∂Ω with respect to the orientation given by N , and N * is the orthogonal projection of N on the tangent bundle of M n . With respect to the functions on ∂Ω to be evaluated in d 2 ds 2 F λ f (0) for a critical point of (VP-1), they have to be considered according to Remark 3, that is, smooth functions on ∂Ω whose integral mean is zero; and, on the other hand, any smooth function on ∂Ω can be evaluated in d 2 ds 2 F λ (0) for a critical point of (VP-2).
Proof: Initially, for any variation X : (− , ) × ∂Ω → M n f × ρ R of ∂Ω we consider the function u 0 ∈ C ∞ (∂Ω) defined in (3.3). Since H f is constant, from (3.10) and (3.9) we have that Reasoning as in the proof of equation (3.5) of [11], we obtain Hence, On the other hand, denoting by N * and N ⊥ the orthogonal projections of N over the tangent and normal bundles of M n , respectively, and taking into account that f is invariant along the flow determined by Y , from [27,Proposition 7.35] we obtain  Therefore, from equations (3.18) and (3.15) we obtain where J f is given in (3.14). Now, for any u ∈ C ∞ (∂Ω), considering variations X : (− , ) × ∂Ω → M n f × ρ R of ∂Ω whose variational field is ∂X ∂t t=0 = uN , we obtain that the last expression (3.19) is also valid for every u ∈ C ∞ (∂Ω). Taking into account the set of functions on ∂Ω that are admissible for a critical point of (VP-2), we conclude that all the arguments stated above are valid to provide the formula of the second variation of F λ f for critical points of (VP-2).
For those critical points of (VP-1), if X : (− , ) × ∂Ω → M n f × ρ R is a variation of ∂Ω which preserve the weighted volume of Ω, then for u 0 ∈ C ∞ (∂Ω) defined in (3.3), we have from Lemma 1 that ∂Ω u 0 dV = 0 and, in addition, the expression (3.19) is valid for such u 0 . Finally, for any function u ∈ C ∞ (∂Ω) such that ∂Ω u dV = 0, from Remark 3 we get a variation X : (− , ) × ∂Ω → M n f × ρ R of ∂Ω which preserves the weighted volume of Ω such that the variational field is ∂X ∂t t=0 = uN , and it follows immediately that (3.19) is retrieved for such a u.
We conclude this subsection by noting that the weighted Jacobi operator J f given in (3.14) belongs to a class of differential operators which are usually referred to as Schrödinger operators, that is, operators of the form ∆ + q, where ∆ is the standard Laplacian on ∂Ω and q is any continuous function on ∂Ω (see, for instance, [18]). In particular, we can highlight that the behavior of the eigenvalues of J f is well known, and this behavior will play an important role in obtaining the main results of this work.

The notion of bifurcation instants for
In what follows, we consider the one-parameter family {Ω γ } γ of open subsets in weighted Killing warped product M n f × ρ R such that the boundary of each Ω γ , denoted by ∂Ω γ , is a closed H f (γ)-hypersurface with constant f -mean curvature H f (γ), where γ varies on a prescribed interval I ⊂ R. In this context, as a consequence of our study of Subsection 3.1, we have that each Ω γ is a critical point of a certain variational problem of type (VP-2). More specifically, each Ω τ is a critical point for the one-parameter family of weighted Jacobi functionals Moreover, from Proposition 2, associated with each closed H 2 (γ)-hypersurface ∂Ω γ we have that the second variation d 2 is the weighted Jacobi operator on ∂Ω γ . Here, ∆ f ;γ and ∆ f are the f -Laplacians on ∂Ω γ and M n f , respectively, Ric f and Hess are the Bakry-Emery-Ricci tensor and the Hessian operator in M n f , A γ is the shape operator of ∂Ω γ with respect to normal vector field N γ , and N * γ is the orthogonal projection of N γ on the tangent bundle of M n .
With respect to our family {Ω γ } γ∈I of critical points of (VP-2), we need to adopt some notions and results that correspond to equivariant bifurcation theory for geometric variational problems. For more details on this subject, we recommend the references [2], [10], [9], and [28].
Let us first recall that two elements Ω γ1 and Ω γ2 of {Ω γ } γ∈I are said to be isometrically congruent when there is an isometry ψ of M n f × ρ R that carries the image of x 1 : ∂Ω γ1 → M n f × ρ R onto the image of , where x 1 and x 2 are the immersions of ∂Ω γ1 and ∂Ω γ2 into M n f × ρ R, respectively, i.e., if there exists a diffeomorphism φ : ∂Ω γ1 → ∂Ω γ2 and an isometry ψ of M n f × ρ R such that the following diagram commutes: Taking into account the studies reported in [9], γ ∈ I is said to be a bifurcation instant for the family {Ω γ } γ∈I if there exists a sequence {γ n } n∈N ⊂ I and a sequence {Ω γn } n∈N ⊂ {Ω γ } γ∈I such that (a) lim n→∞ γ n = γ, x n = x, where x n : Ω γn → M n f × ρ R and x : Ω γ → M n f × ρ R are the immersions of Ω γn and Ω γ into M n f × ρ R, respectively, and (c) for all n ∈ N, x n is not isometrically congruent to x. details, see [4]). Hence, our assumption that d 2 ds 2 F λ(γj ) f (0) is nonsingular for j ∈ {1, 2} is a necessary condition to reach at the bifurcation.
In this paper we will study the local rigidity and the bifurcation instants of {Ω γ } γ∈I by analyzing the spectrum of J f ;γ for all γ ∈ I. Essentially, we will determine the number of negative eigenvalues for each γ (counting its multiplicity) and we will study the evolution of such a number.

Local rigidity and bifurcation instants in
Our first result given in Theorem 1 provides some simple sufficient conditions to get the local rigidity of the family {Ω γ } γ∈I of critical points of the variational problem (VP-2) described in Subsection 3.2.
of ∆ f , with associated eigenvalue c, we have that J f ;γ can be written Remark 8. Let x : Σ n → M n f × ρ R be a closed H f -hypersurface as described in the last definition above. We consider the set Just as in [5], we can establish the following criterion of f -stability: a hypersurface x : In what follows, associated with a hypersurface x : Σ n → M n f × ρ R we will consider a particular smooth function, namely, the angle function where N is the normal vector field on Σ n that determines its orientation and Y is the Killing vector field on M n f × ρ R. In this setting, we get the following key lemma, which provides sufficient conditions to obtain a eigenfunction of the drift Laplacian ∆ f on Σ n . Let us denote by ∇, ∇, and ∇ the Levi-Civita connections of M n f × ρ R, Σ n and M n , respectively.  Our stability result stated in Theorem 3 gives us a characterization of f -stable H f -hypersurfaces in M n f × ρ R through the first eigenvalue of the drift Laplacian ∆ f , which extends a classic result of Barbosa, do Carmo, and Eschenburg (see Proposition 2.13 of [6]).

Proof of Theorem 3:
Since µ is constant, Proposition 3 guarantees that µ is in the spectrum of the drift Laplacian ∆ f . So, let µ 1 be the first eigenvalue of ∆ f on Σ n . If µ = µ 1 , then the variational characterization of λ 1 (see, for instance, Section 1 of [7]) gives where G is defined in (5.1). Then, from (3.13) and (3.14) we obtain for any u ∈ G and, according to Remark 8, x : Σ n → M n f × ρ R is f -stable. Now suppose that x : Σ n → M n f × ρ R is f -stable, which according to Remark 8 is equivalent to d 2 ds 2 F f (0)(u) ≥ 0 for all u ∈ G. Let u be an eigenfunction associated to the first eigenvalue µ 1 of the drift Laplacian ∆ f on Σ n . Consequently, by (3.13) and (3.14) we get Therefore, since µ 1 ≤ µ, we must have µ 1 = µ.