Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 62, pp. 1–10.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.62

COMPLETE NONCOMPACT AND STOCHASTICALLY COMPLETE

m-QUASI YAMABE GRADIENT SOLITONS

GIOVANNI MOLICA BISCI, HENRIQUE F. DE LIMA, ARY V. F. LEITE, MARCO A. L. VELÁSQUEZ

Abstract. We establish new characterization and nonexistence results concerning complete
noncompact and stochastically complete m-quasi Yamabe gradient solitons through the appli-

cations of a key Bochner type formula jointly with suitable maximum principles dealing, in
particular, with the notions of convergence to zero at infinity and polynomial and exponential

volume growth.

1. Introduction

In 1960, using calculus of variations and elliptic partial differential equations techniques, Yam-
abe [27] believed he had solved the following problem (also known as Yamabe problem): Every
compact Riemannian manifold has a conformal metric of constant scalar curvature. However, in
1968 Trudinger [25] discovered an error in Yamabe’s proof. Independently, Trudinger [25] and
Aubin [4] contributed to the Yamabe problem showing that it can be solved on any compact man-
ifold Σn with the additional assumption that the Yamabe invariant of Σn is less than the Yamabe
invariant of the round sphere.

Moreover, Aubin also showed that if Σn has dimension n ≥ 6 and it is not locally conformallly
flat, then the Yamabe invariant of Σn is less than the Yamabe invariant of the round sphere. In
1984, Schoen [23] completed the solution of Yamabe problem showing that if Σn has dimension
n ∈ {3, 4, 5} or if Σn is locally conformally flat, then the Yamabe invariant of Σn is less than the
Yamabe invariant of the round sphere, unless Σ is conformal to the round sphere (for more details,
we recommend [15]).

Afterwards, Hamilton [12] introduced the Yamabe flow of a Riemannian manifold (Σn, g) which
describes how the metric g is deformed to a metric g(t) at time t through the evolution equation

∂g(t)

∂t
= −R(t)g(t)

g(0) = g,
(1.1)

where R(t) stands the scalar curvature related to metric g(t). In this branch, a special solution of
Yamabe flow, called Yamabe soliton, is a self-similar solution of (1.1) such that

1

2
LXg = (R− ρ)g, (1.2)

where LXg denotes the Lie derivative of the metric g with the respect to some smooth vector field
X ∈ X(Σn) and ρ ∈ R is a constant. For the particular case that X = ∇f , where ∇f denotes
the gradient (related to metric g) of a smooth function f ∈ C∞(Σn), it is called gradient Yamabe
soliton and (1.2) becomes

∇2f = (R− ρ)g, (1.3)

2020 Mathematics Subject Classification. 53C25, 53C44.
Key words and phrases. Complete and stochastically complete m-quasi Yamabe gradient solitons;

scalar curvature; convergence to zero at infinity; polynomial and exponential volume growth.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted April 14, 2025. Published June 25, 2025.

1



2 G. MOLICA BISCI, H. F. DE LIMA, A. V. F. LEITE, M. A. L. VELÁSQUEZ EJDE-2025/62

where ∇2f stands for the Hessian (related to metric g) of f . A first generalization of the gradient
Yamabe soliton, introduced by Huang and Li in [14], is called quasi Yamabe gradient soliton and
defined by

∇2f − 1

m
∇f ⊗∇f = (R− ρ)g, (1.4)

for some nonzero m ∈ R. When the function f is constant we say that the quasi Yamabi gradient
soliton is trivial. Moreover, it is not difficult to see that when m → +∞ the equation (1.4) returns
to equation (1.3). In this same paper, Huang and Li showed that a compact quasi Yamabe gradient
soliton has constant scalar curvature, extending a previous result due to Hsu [13].

Regarding to the noncompact case, Wang [26] gave an example of a noncompact quasi Yamabe
gradient soliton with a nonconstant scalar curvature (for more details see [26, Example 2.1]).
Besides, among other results of this work, Wang showed some scalar curvature estimates and,
under a suitable integrability condition, he also proved that a quasi Yamabe gradient soliton has
constant scalar curvature.

In [16], Naik introduced the notion of concurrent-recurrent vector field as a vector field ν which
satisfies the relation

∇Xν = α{X − ν♭(X)ν},
where α ∈ R\{0} and ν♭ is the 1-form equivalent to ν in a Riemannian manifold (Σn, g). Lately,
Naik, Ramandi, Kumara and Venkatesha [17] used the above concept to prove that an n-dimensional
nontrivial quasi Yamabe gradient soliton which admits a concurrent-recurrent vector field has con-
stant scalar curvature equal to −n(n− 1)α2.

More recently, Poddar, Sharma and Subramanian [22] established the concept of m-quasi Yam-
abe gradient soliton. More precisely, let (Σn, g, u) be a Riemannian manifold Σn endowed with the

metric g and a smooth function u = e−
f
m ∈ C∞(Σn), where f ∈ C∞(Σn) is a smooth function on

Σn and 0 < m < +∞ is a real number. Since ∇u = − u
m∇f , with a straightforward computation

we obtain that

∇2u =
u

m2
∇f ⊗∇f − u

m
∇2f, (1.5)

and, from (1.4), we arrive at
1

2
L∇ug = − u

m
(R− ρ)g,

or, equivalently,

∇2u = − 1

m
(R− ρ)ug. (1.6)

When the function u is constant we say that the m-quasi Yamabe gradient soliton is trivial. In
this setting, Poddar, Sharma and Subramanian [22] showed that every compact m-quasi Yamabe
gradient soliton (Σn, g, u), n > 2, has constant scalar curvature.

Proceeding with this picture, in the present paper we extend the techniques of [7] in order
to establish new characterization and nonexistence results concerning complete noncompact and
stochastically complete m-quasi Yamabe gradient solitons (Σn, g, u) through the applications of
a key Bochner type formula jointly with suitable maximum principles dealing, in particular, with
the notions of convergence to zero at infinity and polynomial and exponential volume growth.

2. Suitable Bochner type formula

In this section, we establish a suitable Bochner type formula which will be used to prove some
of our main results. We start remembering the classical Bochner formula (see for instance [5]):

1

2
∆|∇u|2 = Ric(∇u,∇u) + g(∇u,∇∆u) + |∇2u|2, (2.1)

where Ric stands for the Ricci tensor of (Σn, g).
On the other hand, considering a (local) orthonormal frame {E1, · · · , En} on Σn, from (1.6)

we have

|∇2u|2 =
∑
i

|∇2u(Ei)|2 =
∑
i

∣∣ 1
m
(R− ρ)uEi

∣∣2 =
n

m2
(R− ρ)2u2. (2.2)



EJDE-2025/62 COMPLETE m-QUASI YAMABE GRADIENT SOLITONS 3

Moreover, choosing {E1, · · · , En} to be a geodesic frame, we claim that

Ric(∇u,∇u) + g(∇u,∇∆u) =
∑
i

g(∇Ei
∇Ei

∇u,∇u). (2.3)

Indeed, we have that

Ric(∇h,∇h) =
∑
i

g(R(∇h,Ei)∇h,Ei)

=
∑
i

g(∇Ei
∇∇h∇h−∇∇h∇Ei

∇h+∇[∇h,Ei]∇h,Ei).
(2.4)

But, since {E1, · · · , En} is a geodesic frame, we obtain∑
i

g(∇∇h∇Ei
∇h,Ei) =

∑
i

∇h(g(∇Ei
∇h,Ei)) = ∇h(∆h) = g(∇h,∇∆h). (2.5)

Moreover, we also obtain

g(∇Ei∇∇h∇h+∇[∇h,Ei]∇h,Ei)

= Ei(g(∇∇h∇h,Ei))− g(∇∇h∇h,∇Ei
Ei)− g(∇Ei

∇h, [Ei,∇h])

= Ei(g(∇Ei
∇h,∇h))− g(∇Ei

∇h,∇Ei
∇h−∇∇hEi)

= g(∇Ei
∇Ei

∇h,∇h).

(2.6)

Hence, inserting (2.5) and (2.6) into (2.4) we arrive at (2.3). Thus, from (1.6) and (2.3) we obtain

Ric(∇u,∇u) + g(∇u,∇∆u) = − 1

m
g(∇((R− ρ)u),∇u). (2.7)

But, taking the trace in (1.6) we obtain

∆u = − n

m
(R− ρ)u. (2.8)

Consequently, from (2.8) we have

g(∇u,∇∆u) = − n

m
g(∇((R− ρ)u),∇u).

So, we conclude that

− 1

m
g(∇((R− ρ)u),∇u) = − 1

(n− 1)
Ric(∇u,∇u). (2.9)

Hence, from (2.1), (2.7) and (2.9) we reach our suitable Bochner type formula

1

2
∆|∇u|2 = |∇2u|2 − 1

n− 1
Ric(∇u,∇u). (2.10)

3. Main results

This section is devoted to establish our characterization and nonexistence results concerning
complete noncompact and stochastically complete m-quasi Yamabe gradient solitons. For this, we
will apply suitable maximum principles as main analytical tools.

3.1. Via integrability conditions. Yau [28] generalizing a previous result due to Gaffney [9],
established the following version of Stokes’ Theorem on an n-dimensional complete noncompact
Riemannian manifold Σn: If ω ∈ Ωn−1(Σn) is an integrable (n− 1)-differential form on Σn, then
there exists a sequence Bi of domains on Σn such that

Bi ⊂ Bi+1, Σn = ∪i≥1Bi and lim
i→+∞

∫
Bi

dω = 0.

Suppose Σn is oriented by the volume element dΣ, and let Lq(Σn) be the space of Lebesgue
q-integrable functions on Σn, that means

Lq(Σn) :=
{
u ∈ C∞(Σn);

∫
Σ

|u|qdΣ < +∞, 1 ≤ q < +∞
}
.



4 G. MOLICA BISCI, H. F. DE LIMA, A. V. F. LEITE, M. A. L. VELÁSQUEZ EJDE-2025/62

If ω = ιXdΣ is the contraction of dΣ in the direction of a smooth vector field X on Σn, then
Caminha obtained the following consequence of Yau’s result (see [6, Proposition 2.1]).

Lemma 3.1. Let X be a smooth vector field on the n-dimensional complete noncompact oriented
Riemannian manifold (Σn, g), such that divg X does not change sign on (Σn, g). If |X| ∈ L1(Σn),
then divg X = 0.

Now, we are in a position to present our first characterization result related to complete non-
compact m-quasi Yamabe gradient soliton.

Theorem 3.2. Let (Σn, g, u) be a complete noncompact m-quasi Yamabe gradient soliton whose
Ricci tensor satisfies Ric(∇u,∇u) ≤ 0 and such that |∇|∇u|2| ∈ L1(Σn), then R = ρ on Σn.

Proof. Let us take the smooth vector field X = ∇|∇u|2 ∈ X(Σn). In this setting, by hypothesis,
we have that |X| ∈ L1(Σn). Moreover, taking into account that Ric(∇u,∇u) ≤ 0, from (2.10) we
also have that

divg X = ∆|∇u|2 = 2
{
|∇2u|2 − 1

n− 1
Ric(∇u,∇u)

}
≥ 0.

Applying Lemma 3.1 we conclude that divg(X) = 0. In particular, we obtain that |∇2u|2 = 0
and, consequently, from (2.2), we have

n

m2
(R− ρ)2u2 = 0.

Therefore, since u > 0, we obtain that R = ρ. □

From Kato’s inequality we observe that∣∣∇|∇u|2
∣∣ = 2|∇u|

∣∣∇|∇u|
∣∣ ≤ 2|∇u| |∇2u|.

Hence, assuming that |∇u| ∈ L∞(Σn) and |∇2u| ∈ L1(Σn) we can rewrite the Theorem 3.2 as
follows.

Theorem 3.3. Let (Σn, g, u) be a complete noncompact m-quasi Yamabe gradient soliton whose
Ricci tensor satisfies Ric(∇u,∇u) ≤ 0, and such that |∇u| ∈ L∞(Σn) and |∇2u| ∈ L1(Σn), then
R = ρ on Σn.

At this point we recall that a smooth function u ∈ C∞(Σn) on a Riemannian manifold Σn is
called a subharmonic function when ∆u ≥ 0 on Σn. In this setting, we present the next lemma
which is a Liouville type result due to Yau [29] (see also [21]).

Lemma 3.4. Let u be a nonnegative smooth subharmonic function on a complete noncompact
Riemannian manifold Σn. If u ∈ Lq(Σn), for some q > 1, then u is constant.

Next, we will assume that u ∈ Lq(Σn), 1 < q < +∞ and R ≤ ρ. In this way, using Lemma 3.4
we obtain the following theorem.

Theorem 3.5. Let (Σn, g, u) be a complete noncompact m-quasi Yamabe gradient soliton such
that R ≤ ρ and u ∈ Lq(Σn), 1 < q < +∞, then (Σn, g, u) is trivial and R = ρ on Σn.

Proof. Since R ≤ ρ, from (2.8) we have

∆u = − n

m
(R− ρ)u ≥ 0.

Thus, since we are assuming u ∈ Lq(Σn) with 1 < q < +∞, Lemma 3.4 guarantees that u is
constant and, hence,

n

m
(R− ρ)u = 0,

implying that R = ρ. □



EJDE-2025/62 COMPLETE m-QUASI YAMABE GRADIENT SOLITONS 5

3.2. Via volume growth. In this subsection we deal with the notion of volume growth. Let us
consider a (connected oriented) complete Riemannian manifold (Σn, g) and denote by B(p, t) a
geodesic ball centered at p and with radius t. Given a continuous function σ : (0,+∞) → (0,+∞)
we say that Σn has volume growth like σ(t) if there exists p ∈ Σn such that

vol(B(p, t)) = O(σ(t))

as t → ∞, where vol stands the volume, that is,

vol(B(p, t)) =

∫
B(p,t)

dΣ.

With the above concept in mind, we can present the following lemma which corresponds to a
particular case of a more general maximum principle due to Aĺıas, Caminha and Nascimento (see
[2, Theorem 2.1]).

Lemma 3.6. Let (Σn, g) be a complete noncompact Riemannian manifold and let X ∈ X(Σn) be a
bounded smooth vector field on Σn, with |X| ≤ c for some positive constant c ∈ R. Let v ∈ C∞(Σ)
be a smooth function such that g(∇v,X) ≥ 0 and divg X ≥ av on Σn, for some positive constant
a ∈ R.

(i) If (Σn, g) has polynomial volume growth, then v ≤ 0 on Σn.

(ii) If (Σn, g) has exponential volume growth, say like eβt, then v ≤ cβ
a on Σn.

Using this previous lemma, we obtain the following nonexistence result concerning complete
noncompact m-quasi Yamabe gradient soliton.

Theorem 3.7. There is no a complete noncompact m-quasi Yamabe gradient soliton (Σn, g, u)
with polynomial volume growth such that |∇u| ∈ L∞(Σn) and R ≤ ρ−α, for some positive constant
α ∈ R.

Proof. Suppose by contradiction the existence of such a complete m-quasi Yamabe gradient soliton
(Σn, g, u). So, let us take the smooth vector field X = ∇u ∈ X(Σn). We point out that u is a
smooth function such that

g(∇u,X) = g(∇u,∇u) ≥ 0.

Since we are supposing that R ≤ ρ− α, we obtain

divg X = ∆u = − n

m
(R− ρ)u ≥ nα

m
u.

Hence, since Σn is complete noncompact we can apply item (i) of Lemma 3.6 to conclude that
u ≤ 0, leading us to an absurd. □

Next, we will use Bochner type formula (2.10) to obtain the following characterization result.

Theorem 3.8. Let (Σn, g, u) be a complete noncompact m-quasi Yamabe gradient soliton whose
Ricci tensor satisfies Ric(∇u,∇u) ≤ −α|∇u|2, for some positive constant α ∈ R. If (Σn, g) has
polynomial volume growth and |∇u|, |∇2u| ∈ L∞(Σn), then (Σn, g, u) is trivial and R = ρ on Σn.

Proof. Taking the smooth vector field X = ∇|∇u|2 ∈ X(Σn) and the smooth function v = |∇u|2 ∈
C∞(Σn), by using again Kato’s inequality we have that

|X| = |∇|∇u|2| = 2|∇u||∇|∇u|| ≤ 2|∇u||∇2u|.

So, since |∇u|, |∇2u| ∈ L∞(Σn), we see that X is a bounded smooth vector field. Moreover, we
also have that

g(X,∇v) = g(∇|∇u|2,∇|∇u|2) ≥ 0.

Furthermore, since Ric(∇u,∇u) ≤ −α|∇u|2, from (2.10) we infer that

divg X = ∆|∇u|2 ≥ 2α

n− 1
|∇u|2.



6 G. MOLICA BISCI, H. F. DE LIMA, A. V. F. LEITE, M. A. L. VELÁSQUEZ EJDE-2025/62

Hence, since Σn is complete noncompact, we can apply item (i) of Lemma 3.6 to conclude that
|∇u|2 = 0 and, consequently, u must be a positive constant. Thus, we also have

0 =
1

2
∆|∇u|2 ≥ |∇2u|2 =

n

m2
(R− ρ)2u2 ≥ 0,

implying that R = ρ. □

Proceeding, we will deal with complete noncompact m-quasi Yamabe gradient solitons having
exponential volume growth.

Theorem 3.9. Let (Σn, g, u) be a complete noncompact m-quasi Yamabe gradient soliton with
exponential volume growth, say like eβt. If |∇u| ∈ L∞(Σn) and R ≤ ρ − α, for some positive
constant α ∈ R, then

|u|∞ ≤ mβ

nα
|∇u|∞.

Proof. Taking the smooth vector field X = ∇u ∈ X(Σn) and following the same steps of the
proof of Theorem 3.7 we obtain that X is a bounded smooth vector field, g(∇u,X) ≥ 0 and
divg X ≥ nα

m u.
Hence, applying item (ii) of Lemma 3.6 we obtain

|u| ≤ mβ

nα
|∇u|∞.

Therefore, we conclude that

|u|∞ ≤ mβ

nα
|∇u|∞. □

To finish this subsection we will present one more result concerning exponential volume growth
of a complete noncompact m-quasi Yamabe gradient soliton.

Theorem 3.10. Let (Σn, g, u) be a complete noncompact m-quasi Yamabe gradient soliton with
exponential volume growth, say like eβt. If |∇u|, |∇2u| ∈ L∞(Σn) and Ric(∇u,∇u) ≤ −α|∇u|2,
for some positive constant α ∈ R, then

|∇u|∞ ≤ (n− 1)β

2α
|∇2u|∞.

Proof. Taking the smooth vector field X = ∇|∇u|2 ∈ X(Σn) and following the same steps of the
proof of Theorem 3.8 we obtain that X is a bounded smooth vector field, g(∇|∇u|2, X) ≥ 0 and
divg X ≥ 2α

n−1 |∇u|2.
So, applying item (ii) of Lemma 3.6 we obtain

|∇u|2 ≤ (n− 1)β

2α
|∇u|∞|∇2u|∞.

Thus, we conclude that

|∇u|∞ ≤ (n− 1)β

2α
|∇2u|∞. □

3.3. Via stochastic completeness. We recall that a Riemannian manifold (Σn, g) is said to be
stochastically complete if, for some (and, hence, for any) (x, τ) ∈ Σn × (0,+∞), the heat kernel
p(x, y, τ) of the Laplace-Beltrami operator ∆ satisfies the conservation property∫

Σ

p(x, y, τ)dµ(y) = 1. (3.1)

From the probabilistic viewpoint, stochastically completeness is the property of a stochastic process
to have infinite life time. Furthermore, for the Brownian motion on a manifold, the conservation
property (3.1) means that the total probability of the particle to be found in the state space is
constantly equal to one (cf. [8, 10, 11, 24]).

On the other hand, following the terminology introduced by Pigola, Rigoli and Setti in [20],
the Omori-Yau’s maximum principle is said to hold on a (not necessarily complete) n-dimensional



EJDE-2025/62 COMPLETE m-QUASI YAMABE GRADIENT SOLITONS 7

Riemannian manifold (Σn, g) if, for any smooth function u ∈ C2(Σn) with supΣ u < +∞, there
exists a sequence of points (pk) ⊂ Σn satisfying

lim
k

u(pk) = sup
Σ

u, lim
k

|∇u(pk)| = 0 and lim sup
k

∆u(pk) ≤ 0.

In this point of view, the classical result given by Omori and Yau in [18, 29] states that Omori-Yau’s
maximum principle holds on every complete Riemannian manifold with Ricci curvature bounded
from below.

But, as it was also observed by Pigola, Rigoli and Setti in [20], the validity of Omori-Yau’s
maximum principle on Σn does not depend on curvature bounds as would be expected. For
instance, the Omori-Yau’s maximum principle holds on every Riemannian manifolds which is
properly immersed into a Riemannian space form with controlled mean curvature (see [20, Example
1.14]). In particular, it holds for every constant mean curvature hypersurface properly immersed
into a Riemannian space form.

More generally, following again the terminology introduced in [20], the weak Omori-Yau’s maxi-
mum principle is said to hold on a (not necessarily complete) n-dimensional Riemannian manifold
(Σn, g) if, for any smooth function u ∈ C2(Σn) with supΣ u < +∞, there exists a sequence of
points (pk) ⊂ Σn with the properties

lim
k

u(pk) = sup
Σ

u and lim sup
k

∆u(pk) ≤ 0.

In this setting, Pigola, Rigoli and Setti [19, 20] proved the following equivalence:

Lemma 3.11. A Riemannian manifold (Σn, g) is stochastically complete if and only if the weak
Omori-Yau’s maximum principle holds on (Σn, g).

With this previous discussion in mind, we are able to present our next characterization result.

Theorem 3.12. Let (Σn, g, u) be a stochastically complete m-quasi Yamabe gradient soliton whose
Ricci tensor satisfies Ric(∇u,∇u) ≤ −α|∇u|2, for some positive constant α ∈ R. If |∇u| ∈ L∞(Σ),
then (Σn, g, u) is trivial and R = ρ on Σn.

Proof. Initially, we recall that

1

2
∆|∇u|2 = |∇u|∆|∇u|+ g(∇|∇u|,∇|∇u|).

Now, applying once more Kato’s inequality, from (2.10) we obtain

|∇u|∆|∇u|+ g(∇|∇u|,∇|∇u|) = |∇2u|2 − 1

n− 1
Ric(∇u,∇u)

≥ |∇|∇u||2 − 1

n− 1
Ric(∇u,∇u).

Since |∇|∇u||2 = g(∇|∇u|,∇|∇u|), from above inequality we obtain

|∇u|∆|∇u| ≥ − 1

n− 1
Ric(∇u,∇u) ≥ α

n− 1
|∇u|2,

where it was used the hypothesis on Ric(∇u,∇u) in the last inequality.
Supposing that supΣ |∇u| > 0 and taking into account that |∇u| ∈ L∞(Σn) we can apply the

Lemma 3.11 to obtain

0 ≥ lim sup
k

∆|∇u| ≥ α

n− 1
sup
Σ

|∇u| > 0,

which is a contradiction. So, we conclude that supΣ |∇u| = 0 and, therefore, u is constant. To
finish the proof we observe that (2.2) gives

n

m2
(R− ρ)2u2 = 0

and, consequently, R = ρ. □



8 G. MOLICA BISCI, H. F. DE LIMA, A. V. F. LEITE, M. A. L. VELÁSQUEZ EJDE-2025/62

We recall that a (non necessarily complete) Riemannian manifold (Σn, g) is called parabolic
when the only subharmonic functions u ∈ C∞(Σn) which are bounded from above are the constant
ones. Taking into account [11, Corollary 6.4], we have that every parabolic Riemannian manifold
is stochastically complete. In this setting, we obtain the following consequence of Theorem 3.12.

Corollary 3.13. Let (Σn, g, u) be a parabolic m-quasi Yamabe gradient soliton whose Ricci tensor
satisfies Ric(∇u,∇u) ≤ −α|∇u|2, for some positive constant α ∈ R. If |∇u| ∈ L∞(Σ), then
(Σn, g, u) is trivial and R = ρ on Σn.

Considering [3, Theorem 2.13] we can establish our second corollary of Theorem 3.12.

Corollary 3.14. Let (Σn, g, u) be a complete m-quasi Yamabe gradient soliton whose Ricci tensor
satisfies Ric(∇u,∇u) ≤ −α|∇u|2, for some positive constant α ∈ R and Ric ≥ −G(r), where
r denotes the Riemannian distance function from a fixed origin in Σn and the function G ∈
C1([0,+∞)) satisfies

G(0) > 0, G′(0) ≥ 0 and G− 1
2 /∈ L1([0,+∞)).

If |∇u| ∈ L∞(Σ), then (Σn, g, u) is trivial and R = ρ on Σn.

3.4. Via convergence at infinity. For our last result, we need the concept to convergence to
zero at infinity. Given a (connected) complete noncompact Riemannian manifold (Σn, g) and
denoting by d(·, o) : Σn → [0,+∞) the Riemannian distance of Σn measured from a fixed point
o ∈ Σn, a function h ∈ C∞(Σn) converges to zero at infinity when

lim
d(x,o)→∞

h(x) = 0.

The following lemma is due to Aĺıas, Caminha and do Nascimento [1].

Lemma 3.15. Let (Σn, g) be a complete noncompact Riemannian manifold and let X ∈ X(Σn)
be a smooth vector field on Σn. Assume that there exists a nonnegative, non-identically vanishing
function v ∈ C∞(M) which converges to zero at infinity and such that g(∇v,X) ≥ 0. If divg X ≥ 0
on Σn, then g(∇v,X) ≡ 0 on Σn.

We close our paper with the following characterization result.

Theorem 3.16. Let (Σn, g, u) be a complete noncompact m-quasi Yamabe gradient soliton whose
Ricci tensor satisfies Ric(∇u,∇u) ≤ 0. If |∇u| converges to zero at infinity, then (Σn, g, u) is
trivial and R = ρ on Σn.

Proof. Suppose by contradiction that (Σn, g, u) is not a trivial m-quasi Yamabe gradient soliton.
Since we are supposing that Ric(∇u,∇u) ≤ 0, taking the smooth vector fieldX = ∇|∇u|2 ∈ X(Σn)
we have that divg X = ∆|∇u|2 ≥ 0. Furthermore, taking v := |∇u|2 we observe that v is a
nonnegative and non-identically vanishing smooth function of C∞(Σn).

On the other hand, we obtain

g(X,∇v) = g(∇|∇u|2,∇|∇u|2) ≥ 0.

So, applying Lemma 3.15 we conclude that g(∇|∇u|2,∇|∇u|2) = 0 and, consequently, |∇u| is
constant. Since |∇u| converges to zero at infinity, we have that |∇u| = 0 and, therefore, u is
constant, which is an absurd.

In this picture, we have verified that (Σn, g, u) must be a trivial m-quasi Yamabe gradient
soliton. In particular, as before, we have

0 = ∆|∇u|2 ≥ n

m2
(R− ρ)2u2 ≥ 0.

Hence, since u > 0, we also conclude that R = ρ. □

Remark 3.17. Let us consider the Euclidean subset

Σn := {(x1, . . . , xn) ∈ Rn;x1 + · · ·+ xn > 0}
endowed with the Euclidean metric gij = δij and potential function f defined by

f(x1, . . . , xn) = −m log(x1 + · · ·+ xn).



EJDE-2025/62 COMPLETE m-QUASI YAMABE GRADIENT SOLITONS 9

We have that (Σn, g) is Ricci-flat and ∇2f = 1
m∇f ⊗ ∇f . In particular, taking u = e−

f
m , from

(1.5) we deduce ∇2u ≡ 0. Hence, from (1.6) we have that (Σn, g, u) is a non-trivial noncompact
stochastically complete m-quasi Yamabe gradient soliton with Ric ≡ 0 and such that ρ = R = 0.
Therefore, through this example, we see the importance of the hypotheses used to establish our
triviality results.

Acknowledgements. This work was funded by the European Union - NextGenerationEU within
the framework of PNRR Mission 4 - Component 2 - Investment 1.1 under the Italian Ministry of
University and Research (MUR) program PRIN 2022 - grant number 2022BCFHN2 - Advanced
theoretical aspects in PDEs and their applications - CUP: H53D23001960006. G. Molica Bisci
was supported by INdAM-GNAMPA Research Project 2024: Aspetti geometrici e analitici di
alcuni problemi locali e non-locali in mancanza di compattezza - CUP E53C23001670001. H. F.
de Lima and A. L. Velásquez were partially supported by CNPq, Brazil, grants 305608/2023-1
and 304891/2021-5, respectively. A.V. F. Leite was supported by CAPES, Brazil.

The authors would like to thank the anonymous referee for his/her valuable suggestions and
useful comments which improved the paper.

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Giovanni Molica Bisci

Department of Human Sciences and Quality of Life Promotion, San Raffaele University of Rome, Rome,

Italy
Email address: giovanni.molicabisci@uniroma5.it

Henrique F. de Lima

Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande,
Paráıba, Brazil

Email address: henriquedelima74@gmail.com

Ary V. F. Leite

Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande,

Paráıba, Brazil
Email address: ary.v.l.f@gmail.com

Marco A. L. Velásquez
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande,

Paráıba, Brazil

Email address: marcolazarovelasquez@gmail.com


	1. Introduction
	2. Suitable Bochner type formula
	3. Main results
	3.1. Via integrability conditions
	3.2. Via volume growth
	3.3. Via stochastic completeness
	3.4. Via convergence at infinity
	Acknowledgements

	References