Main Results

This section is devoted to establish our uniqueness results concerning two-sided hypersurfaces in a weighted product space, which will be obtained using the maximum principles quoted in the Section 1.1. Now, we present our first one.

Theorem 3.1. Let Mⁿ⁺¹_f = R× M_fⁿ be a weighted product space and let ψ : Σⁿ → Mⁿ⁺¹_f be a complete two-sided hypersurface, with Bakry- ´Emery-Ricci tensor satisfying Ric_f ≥α⟨ , ⟩, for some positive constantα. If the f-mean curvature H_f is a nonpositive (nonnegative) constant and the Weingarten operatorAis positive (negative) semi-definite, then Σⁿ must be a slice.

Proof. From the Bochner-Lichnerowicz’s formula for the drift Laplacian (see [70]), we have that

1

2∆_f|∇h|² =|Hessh|²+ Ric_f(∇h,∇h) +⟨∇h,∇(∆_fh)⟩. (3.13) But, taking into account our constraints on H_f and A, from (3.8) and (3.12) we get

⟨∇h,∇(∆_fh)⟩=−nH_f⟨∇h, A∇h⟩ ≥0. (3.14) Moreover, from our assumption on Ric_f,

Ric_f(∇h,∇h)≥α|∇h|². (3.15) Hence, using (3.14) and (3.15) into (3.13), we obtain

1

2∆f|∇h|² ≥α|∇h|² ≥α|∇h|⁴.

Therefore, we can apply Proposition 1.2 to conclude that |∇h| must be vanishes on Σⁿ which means that h is constant on it, that is, Σⁿ is a slice.

In particular, from Theorem 3.1 we obtain

Corollary 3.2. The only complete two-sided f-minimal hypersurfaces immersed in a weighted product space R × M_fⁿ and with Bakry- ´Emery-Ricci tensor Ric_f satisfying Ric_f ≥α⟨ , ⟩, for some positive constant α, are the slices.

We say that hypersurface Σⁿimmersed inR×M_fⁿlies in a half-spaceif Σⁿis contained in a region of R×M_fⁿ of form [τ,+∞)×M_fⁿ or (−∞, τ]×M_fⁿ, for some τ ∈R. In this setting, we obtain the following result:

Theorem 3.3. The only complete two-sidedf-minimal hypersurfaces lying in a half-space of the weighted product space R×M_fⁿ and whose Bakry- ´Emery-Ricci tensor is bounded from below, are the slices.

Proof. Let Σⁿ be such a complete two-sided f-minimal hypersurface lying, for instance, in a half-space [τ,+∞)×M_fⁿ, someτ ∈R. In this case, the height function of Σⁿsatisfies h−τ ≥0. So, we consider on Σⁿ the smooth function

F := 1 1 +h−τ.

We have that 0< F ≤1 and, for any X∈X(Σ),

⟨∇F, X⟩=X

1 1 +h−τ

= −X(h)

(1 +h−τ)² = −⟨∇h, X⟩

(1 +h−τ)², implying that

∇F =−F²∇h. (3.16)

Thus, from (3.3), (1.8) and (3.16) we get

∆_fF = ∆F +F²⟨∇f, ∂_t^⊤⟩. (3.17) Since we are assuming that f does not depend on the parameter t ∈ R, that is,

⟨∇f, ∂_t⟩= 0, writing∂_t=∂_t^⊤+ηN, we have that ⟨∇f, ∂_t^⊤⟩=−⟨∇f, N⟩η. Consequently, from (3.17) we reach at

∆_fF = ∆F −F²⟨∇f, N⟩η. (3.18) But, using the property div(φY) =Y(φ) +φdiv(Y), which holds for any smooth vector field Y and any smooth function φon Σⁿ, we have that

∆F = div(∇F) = ∇h(−F²)−F²∆h= 2F³|∇h|²−F²∆h. (3.19) Moreover, since Σⁿ is f-minimal, from (1.8) and (3.12) we get

∆h=−⟨∇f, N⟩η. (3.20)

Hence, from (3.18), (3.19) and (3.20) we obtain

∆_fF = 2F³|∇h|² ≥0. (3.21)

Since Σⁿ has Bakry-´Emery-Ricci tensor bounded from below and F is (in particular) bounded from above, from the weak Omori-Yau’s maximum principle (see, for instance, [63, Remark 2.18] or [18, Theorem 5.4]), there exists a sequence of points {x_k}k∈N ⊂Σⁿ such that

limk F(xk) = sup

Σⁿ

F and lim sup

k

∆fF(xk)≤0. (3.22) Therefore, from (3.21) and (3.22) we have that ∆fF vanishes identically on Σⁿ and, returning to (3.21), we conclude that |∇h|² is identically zero, which means that Σⁿ is a slice.

When Σⁿ lies in a half-space of the type (−∞, τ]×M_fⁿ, we can consider the smooth function

0<F˜ := 1

1 +τ−h ≤1

and, using once more the weak Omori-Yau’s maximum principle, we obtain the same conclusion.

In our next result, we will assume that a lower boundedness on the Bakry-´Emery-Ricci tensor of the base of the ambient space. The following result generalize the Theorem 6 of [22] without the constraints of limitation of the Weingarten operator and the angle function.

Theorem 3.4. LetMⁿ⁺¹_f =R×M_fⁿbe a weighted product space whose Bakry- ´Emery-Ricci tensor Ricf_f of the base (Mⁿ, g) satisfies Ricf_f ≥ αg, for some positive constant α ∈ R. If ψ : Σⁿ → Mⁿ⁺¹_f is a complete two-sided hypersurface with Bakry- ´Emery-Ricci tensor bounded from below, constant f-mean curvature and height function satisfying|∇h|² ≤ ¹₂, then Σⁿ is a slice.

Proof. From [20, Proposition 2.1], we have that

∆η =−(Ric(Nf ^∗, N^∗) +|A|²)η−n⟨∇H, ∂_t⟩, (3.23) whereRic denotes the Ricci tensor of (Mf ⁿ, g) andN^∗ =N−η∂_tstands for the orthogonal projection ofN onto Mⁿ. Thus, since H_f is constant, from (3.4) and (3.23) we get

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2∆_f|∇h|² =−η∆η− |∇η|²+⟨∇f,∇η⟩η

= (Ric(Nf ^∗, N^∗) +|A|²)η²+n⟨∇H, ∂_t⟩η− |∇η|²+⟨∇f,∇η⟩η

= (Ric(Nf ^∗, N^∗) +|A|²)η²−∂_t^⊤(⟨∇f, N⟩)η− |∇η|²+⟨∇f,∇η⟩η. (3.24) On the other hand, using the decomposition ∂_t^⊤ =∂_t−ηN, we obtain

∂_t^⊤(⟨∇f, N⟩) = ⟨∇_∂^⊤

t ∇f, N⟩+⟨∇f,∇_∂^⊤

t N⟩

=⟨∇_∂t∇f, N⟩ −η⟨∇_N∇f, N⟩+⟨∇f,∇η⟩.

But, since f does not depend on the parameter t∈R, we have

⟨∇_∂t∇f, N⟩=⟨∇_N∇f, ∂_t⟩=N⟨∇f, ∂_t⟩ − ⟨∇f,∇_N∂_t⟩= 0.

Consequently,

−∂_t^⊤(⟨∇f, N⟩)η= Hessf(N, N)η²− ⟨∇f,∇η⟩η. (3.25) We also have that

Hessf(N, N) = Hessf(N^∗, N^∗) + 2ηHessf(N^∗, ∂_t) +η²Hessf(∂_t, ∂_t).

So, observing that Hessf(N^∗, ∂t) = Hessf(∂t, ∂t) = 0, we get

Hessf(N, N) = Hessf(N^∗, N^∗). (3.26) Thus, using (3.5), (3.8), (3.24), (3.25), (3.26) and taking into account our constraints onRicff and |∇h|², we obtain

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2∆f|∇h|² ≥(Ric(Nf ^∗, N^∗) +|A|²)η²+ Hessf(N^∗, N^∗)η²− |A|²|∇h|²

≥Ricf_f(N^∗, N^∗)η²+|A|²(η²− |∇h|²)

≥α|N^∗|²η²

≥α|∇h|⁴.

Therefore, Proposition 1.2 guarantees that |∇h|² must vanishes on Σⁿ, that is, Σⁿ is a slice.

From Theorem 3.4 we derive the following result.

Corollary 3.5. Let M_fⁿ be a shrinking gradient Ricci soliton and let ψ : Σⁿ → R×M_fⁿ be a complete two-sided hypersurface whose Bakry- ´Emery-Ricci tensor is bounded from below. If Σⁿ has constant f-mean curvature and its height function satisfies |∇h|² ≤ ¹₂, then Σⁿ is a slice.

Taking into account [53, Lemma 2], Theorem 3.4 also gives the following consequence.

Corollary 3.6. Let Mⁿ⁺¹_f =R×M_fⁿ be a weighted product space such that the base Mⁿ has nonnegative sectional curvature and Hessf is bounded from below. The only complete two-sided f-minimal hypersurfaces immersed inMⁿ⁺¹_f , with bounded Weingarten operator and whose height function satisfies |∇h|² ≤ ₂¹, are the slices.

In our next result, we will assume that the Bakry-´Emery-Ricci tensor Ric_f of the hypersurface Σⁿ is positive in the direction of the gradient of its height function. For this, we will consider Ric_f(X) := Ric_f(X, X), forX ∈X(Σ).

Theorem 3.7. Let Mⁿ⁺¹_f =R×M_fⁿ be a weighted product space with bounded potential function f and let ψ : Σⁿ →Mⁿ⁺¹_f be a complete noncompact two-sided hypersurface with polynomial f-volume growth and whose Bakry- ´Emery-Ricci tensor satisfies Ric_f(∇h) ≥ α|∇h|², for some positive constant α ∈ R. If the f-mean curvature H_f of Σⁿ is a nonpositive (nonnegative) constant and that its Weingarten operator A is bounded and positive (negative) semi-definite, then Σⁿ must be a slice.

Proof. Taking into account our constraints on H_f and A, we have that

−nHf⟨∇h, A∇h⟩ ≥0.

Thus, since we are assuming that Ric_f(∇h)≥ α|∇h|², from the Bochner-Lichnerowicz’s formula (3.13) jointly with (3.14) we obtain

∆_f|∇h|² ≥2α|∇h|². (3.27)

Moreover, from (3.8) and the boundedness of A we get

|∇|∇h|²|= 2|η||A(∇h)| ≤2|A||∇h|<+∞.

Hence, since Σⁿ has polynomial f-volume growth and f is bounded, we have that Σⁿ has polynomial volume growth. So, we can apply Lemma 1.4 to conclude that |∇h| is identically zero, which means that Σⁿ must be a slice.

Proceeding, we apply Lemma 1.5 to establish the following uniqueness result.

Theorem 3.8. Let Mⁿ⁺¹_f = R × M_fⁿ be a weighted product space with nonnegative potential function f and let ψ : Σⁿ → Mⁿ⁺¹_f be a complete noncompact two-sided hypersurface with polynomial volume growth and whose Bakry- ´Emery-Ricci tensor satisfies Ric_f(∇h) ≥ αe^f|∇h|², for some positive constant α ∈ R. If the f-mean curvature H_f of Σⁿ is a nonpositive (nonnegative) constant and the Weingarten operator A is bounded and positive (negative) semi-definite, then Σⁿ must be a slice.

Proof. The Bochner-Lichnerowicz’s formula (3.13) gives

∆f|∇h|² ≥2αe^f|∇h|².

Moreover, since f is nonnegative and A is bounded, we obtain

|∇|∇h|²|= 2|η||A(∇h)| ≤2e^f|A||∇h| ≤δe^f, for some positive constant δ∈R.

Therefore, since we are also assuming that Σⁿ has polynomial volume growth, we can apply Lemma 1.5 to conclude that|∇h| vanishes identically on Σⁿ and, hence, it must be a slice.

Next, we use Lemma 1.7 to establish other uniqueness result.

Theorem 3.9. Let Mⁿ⁺¹_f = R × M_fⁿ be a weighted product space whose base Mⁿ is complete noncompact and let ψ : Σⁿ → Mⁿ⁺¹_f be a complete noncompact two-sided hypersurface whose Bakry- ´Emery-Ricci tensor satisfies Ric_f(∇h) ≥ α|∇h|², for some positive constant α. Suppose that the f-mean curvature H_f of Σⁿ is a nonpositive (nonnegative) constant and that the Weingarten operator A is positive (negative) semi-definite. If |∇h| converges to zero at infinity, then Σⁿ must be a slice.

Proof. Let us suppose by contradiction that Σⁿ is not a slice or, equivalently, that |∇h|

does not vanish identically on it. Taking X = e^−f∇|∇h|², from our hypotheses jointly with (1.7), (1.8) and (3.27) we obtain that

divX =e^−f∆_f|∇h|² ≥0.

Moreover, choosing u=|∇h|² we also have that

⟨∇u, X⟩=e^−f|∇|∇h|²|² ≥0. (3.28) Consequently, since we are also supposing that |∇h| converges to zero at infinity, we can apply Lemma 1.7 to get that ⟨∇u, X⟩ ≡ 0 on Σⁿ. So, returning to (3.28) we conclude that|∇h|must be constant on Σⁿ. Therefore, from (1.25) we have that|∇h|is identically zero on Σⁿ and, hence, we reach at a contradiction.

We close this section quoting the following consequence of Theorem 3.9.

Corollary 3.10. LetMⁿ⁺¹_f =R×M_fⁿ be a weighted Lorentzian product space whose base Mⁿ is complete noncompact and letψ : Σⁿ→Mⁿ⁺¹_f be a complete noncompact two-sided f-minimal hypersurface whose Bakry- ´Emery-Ricci tensor satisfies Ricf(∇h)≥0. If |∇h|

converges to zero at infinity, then Σⁿ must be a slice.