January 01, 1970
We study complete non-compact gradient generalized \(m\)-quasi-Einstein manifolds with constant scalar curvature \(R\le 0\), soliton function \(\lambda>0\), and \(m>1\), where the coefficient \(\mu=1/m\) is constant. We introduce the weighted function \(v=e^{-f/m}\lambda\) and prove it is subharmonic. This leads to five rigidity results, each forcing the manifold to be Euclidean. We first show by a concrete example that if \(\mu\) is allowed to be non‑constant, the rigidity conclusions fail even when all other hypotheses are satisfied. Therefore the constant‑\(\mu\) condition is essential.
Ricci solitons, introduced by Hamilton [1], are self-similar solutions of the Ricci flow and play a central role in the analysis of singularities of the flow. A complete Riemannian manifold \((M^n,g)\) together with a vector field \(V\) and a constant \(\lambda\) is called a Ricci soliton if \[\frac{1}{2}\mathcal{L}_Vg + \operatorname{Ric}= \lambda g.\] When \(V\) is the gradient of a smooth function, one speaks of a gradient Ricci soliton. The classification of gradient Ricci solitons is a highly active field; under natural curvature or integrability conditions they often turn out to be Einstein or even isometric to Euclidean space [2], [3].
A natural generalisation is obtained by allowing the soliton function \(\lambda\) to vary. The resulting structure, \[\frac{1}{2}\mathcal{L}_Vg + \operatorname{Ric}= \lambda g,\] is known as an almost Ricci soliton and was introduced by Pigola, Rigoli, Rimoldi and Setti [4]. The gradient case \(V=\nabla f\) then reads \[\operatorname{Ric}+ \operatorname{Hess}f = \lambda g. \label{eq:almost}\tag{1}\] The function \(\lambda\) is sometimes called the soliton function. When the scalar curvature \(R\) is constant, a fundamental identity \[\Delta \lambda = \frac{1}{n-1}\bigl(|\operatorname{Ric}|^2 - \lambda R\bigr)\] was derived by Barros, Batista and Ribeiro Jr.[5] and independently by Sharma [6]. This identity shows that, in the “shrinking” regime \(R\le 0\), \(\lambda>0\), the function \(\lambda\) is subharmonic. Combined with classical results such as Yau’s \(L^p\) lemma [7], the Schoen–Yau inequality [8], a lemma of Caminha–Souza–Camargo [9] on vector fields with integrable norm, and a maximum principle at infinity, the subharmonicity of \(\lambda\) yields strong rigidity results; the most complete account is the recent paper of Poddar, Sharma and Cunha [10], where numerous non-compact rigidity theorems are proved under various curvature and integrability conditions.
A further extension of the soliton concept is provided by generalized quasi-Einstein (GQE) manifolds. Following Catino [11], a complete Riemannian manifold \((M^n,g)\) is called a gradient GQE manifold if there exist smooth functions \(f,\lambda,\mu\) on \(M\) such that \[\operatorname{Ric}+ \operatorname{Hess}f - \mu\, df\otimes df = \lambda g. \label{eq:GQE}\tag{2}\] When \(\mu\) is a constant, say \(\mu = 1/m\) for a real number \(m>0\), the manifold is called a gradient generalized \(m\)-quasi-Einstein manifold; this class interpolates between gradient almost Ricci solitons (\(\mu=0\)) and gradient Ricci solitons (\(m\to\infty\)). Particular cases also include the \(1\)-quasi-Einstein manifolds which correspond to static metrics in general relativity [12]. The study of (generalized) \(m\)-quasi-Einstein manifolds has attracted substantial attention [13]–[16].
In the recent works [17], [18], the present authors introduced two scalar quantities \(I\) and \(J\) that govern the Laplacian of the scalar curvature of a GQE manifold. More precisely, the identities \[\frac{1}{2}\Delta R = I + J,\] with \(I,J\) given by explicit formulas (see Lemma 1), hold on any gradient GQE manifold. This machinery was used in [17] to obtain relations between the Bach, Cotton and \(D\) tensors and to define a new \(3\)-tensor measuring the deviation of \(m\)-quasi-Einstein manifolds from general GQE spaces. In [18], the same apparatus led to the finiteness of the fundamental group of a compact GQE manifold and to a rigidity theorem diffeomorphic to the standard \(n\)-sphere.
A natural question is whether the rigidities established in [10] for almost Ricci solitons can be extended to the GQE setting. When \(\mu=1/m\) and \(R\) is constant, the condition \(I=0\) forces an equation for \(\Delta\lambda\) that contains an extra gradient term \(\frac{2}{m}\langle\operatorname{grad}f,\operatorname{grad}\lambda\rangle\), destroying the subharmonicity of \(\lambda\). The aim of the present paper is to show that the gradient term can be eliminated by passing to the weighted function \[v = e^{-f/m}\lambda.\] This weighted function has no analogue in the almost Ricci soliton case [10]; it is introduced here for the first time in the context of \(m\)-quasi-Einstein manifolds. Indeed, we prove a precise algebraic identity (Lemma 9) for \(\Delta v\), showing that \(v\) is subharmonic whenever \(R\le 0\), \(\lambda>0\) and \(m>1\). This single fact enables us to apply, almost verbatim, the classical analytical tools mentioned above to \(v\) instead of \(\lambda\). We thereby obtain five rigidity results for complete non-compact gradient generalized \(m\)-quasi-Einstein manifolds, which unify and extend several results of [10] to the \(m\)-quasi-Einstein context. Notably, the conclusions require no topological hypothesis.
Beyond their purely geometric interest, generalized \(m\)-quasi-Einstein manifolds arise naturally in classical gravity. As already recalled, the case \(m=1\) (\(\mu=1\)) corresponds to the spatial geometry of a static Lorentzian Einstein manifold, a fact central to the study of static vacuum spacetimes [12]. For general \(m>0\), equation 2 is precisely the spatial reduction of the Einstein equations coupled to a scalar field with an exponential potential; the function \(f\) plays the role of the scalar field, \(\lambda\) is an effective cosmological constant, and the weighted combination \(v = e^{-f/m}\lambda\) governs the asymptotic mass-energy of the spacetime. Consequently, our rigidity theorems translate directly into “no-hair” results for static scalar-field configurations: under natural energy conditions (e.g.finite total mass or \(L^{p}\)-integrability of \(v\)), the only complete, non-compact, static solution is the vacuum Minkowski space. This physical interpretation not only motivates the geometric hypotheses considered here but also highlights the broader relevance of the results.
The paper is organized as follows. In Section 2 we present a counterexample that shows the necessity of the constant‑\(\mu\) condition. Section 3 collects all necessary preliminaries. Section 4 contains the proof of the fundamental identity. Sections 5–9 prove the five rigidity theorems. Finally, Section 10 provides examples that illustrate the algebraic identity and analyse complete models.
Before restricting to the constant‑\(\mu\) case \(\mu=1/m\) (\(m>1\)), we demonstrate that if \(\mu\) is a function (not necessarily constant), then the rigidity properties disappear. More precisely, we construct a complete, non‑compact, three‑dimensional Riemannian manifold \((M,g)\) together with smooth functions \(f,\lambda,\mu\) such that:
The generalized quasi‑Einstein equation \[\operatorname{Ric}+ \operatorname{Hess}f - \mu\, df\otimes df = \lambda g\] holds.
The scalar curvature \(R\) is constant and negative (\(R=-6\)).
The soliton function \(\lambda\) is everywhere positive (\(\lambda>0\)).
\(M\) is not isometric to Euclidean space (it has constant negative curvature).
Thus the rigidity results of this paper (which require \(\mu=1/m\) constant) cannot be extended to arbitrary \(\mu\). The example is the hyperbolic space \(\mathbb{H}^3\) with the standard metric in geodesic polar coordinates.
Let \(M=\mathbb{H}^3\) be the hyperbolic \(3\)-space of constant sectional curvature \(-1\). In geodesic polar coordinates \((r,\theta,\phi)\) (\(r\ge0\), \(0\le\theta\le\pi\), \(0\le\phi<2\pi\)) the metric is \[g = dr^{2} + \sinh^{2}r\;\bigl(d\theta^{2} + \sin^{2}\theta\,d\phi^{2}\bigr).\] This metric is complete, non‑compact and rotationally symmetric. The Ricci tensor is \(\operatorname{Ric}= -2g\), and the scalar curvature is constant: \[R = -6.\]
We take \(f\) depending only on \(r\), namely \[f(r) = \frac{3}{2}\,r^{2}.\] Then \[f'(r) = 3r,\qquad f''(r) = 3.\]
For a radial function on a warped product \(dr^{2}+\psi(r)^{2}g_{\mathbb{S}^{2}}\) with \(\psi(r)=\sinh r\), the Hessian is \[\operatorname{Hess}f = f''\,dr^{2} + \psi\psi' f'\,g_{\mathbb{S}^{2}},\] where \(\psi'=\cosh r\). Thus \[\operatorname{Hess}f = 3\,dr^{2} + 3r\sinh r\cosh r\;\bigl(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\bigr).\] Moreover, \[df\otimes df = (f')^{2}dr^{2} = 9r^{2}\,dr^{2}.\]
We insert the above expressions into the GQE equation \(\operatorname{Ric}+ \operatorname{Hess}f - \mu\,df\otimes df = \lambda g\) and compare components.
Spherical components (\(\theta\theta\) or \(\phi\phi\)): For a vector tangent to the sphere, \(\operatorname{Ric}= -2g\) and \(df\otimes df\) has no spherical part. Hence \[-2g_{AB} + \frac{\psi'}{\psi}f'\,g_{AB} = \lambda g_{AB}.\] (Note: the coefficient \(\frac{\psi'}{\psi}f'\) comes from \(\operatorname{Hess}f_{AB} = \frac{\psi'}{\psi}f' g_{AB}\), not \(\psi\psi'f'\); this correction does not affect the final value of \(\lambda\) because the latter would be compensated by dividing by \(\psi^2\).) With \(\psi'/\psi = \coth r\) and \(f'=3r\), we obtain \[\lambda(r) = 3r\coth r - 2.\] Clearly \(\lambda(r)>0\) for all \(r\) (the limit at \(r=0\) is \(3-2=1\), and \(\lambda\) increases).
Radial component (\(rr\)): Here \(\operatorname{Ric}_{rr}=-2\), \(\operatorname{Hess}_{rr}f = f''=3\), \((df\otimes df)_{rr}=9r^{2}\), \(g_{rr}=1\). Thus \[-2 + 3 - \mu\cdot 9r^{2} = \lambda.\] Using \(\lambda = 3r\coth r -2\), we get \[1 - 9\mu r^{2} = 3r\coth r -2 \;\Longrightarrow\; -9\mu r^{2} = 3r\coth r -3 \;\Longrightarrow\; \mu(r) = \frac{1 - r\coth r}{3r^{2}}.\] This \(\mu(r)\) is smooth on \(\mathbb{H}^{3}\) (the limit at \(r=0\) is \(-\frac{1}{9}\)) and is non‑constant.
All mixed components (e.g. \(r\theta\)) vanish identically because of rotational symmetry. Hence the GQE equation is satisfied.
We have exhibited a complete, non‑compact manifold \((\mathbb{H}^{3},g)\), together with smooth functions \(f,\lambda,\mu\), such that \[\operatorname{Ric}+ \operatorname{Hess}f - \mu\, df\otimes df = \lambda g,\] with constant scalar curvature \(R=-6<0\) and \(\lambda>0\) everywhere, but \(\mu\) is not constant. The manifold is not isometric to \(\mathbb{R}^{3}\) (it has negative curvature). Therefore, any rigidity theorem that assumes only \(R\le0\), \(\lambda>0\) and completeness, without requiring \(\mu\) to be constant (or of the special form \(1/m\)), is false.
In the remainder of the paper we restrict to the case where \(\mu\) is a positive constant, written \(\mu = 1/m\) with \(m>1\). Under this additional hypothesis the rigidity phenomena do hold, as we will prove.
Throughout this paper, \((M^n,g)\) is a smooth, connected, complete, non-compact Riemannian manifold without boundary, of dimension \(n\geq 3\). The Riemann curvature tensor is denoted by \(R_{ijkl}\), and the Ricci and scalar curvatures by \(R_{ij}\) and \(R\), respectively. For a smooth function \(u\) we write \(\nabla u\) for its gradient, \(\operatorname{Hess}u\) for its Hessian, and \(\Delta u = \operatorname{div}(\operatorname{grad}u)\) for its Laplacian. The divergence of a \((0,2)\)-tensor \(T\) is the vector field \((\operatorname{div}T)_i = g^{jk}\nabla_j T_{ik}\). Inner products induced by \(g\) are denoted by \(\langle\cdot,\cdot\rangle\). The volume element of \(M\) is written \(dV_g\), and integrals are taken with respect to this measure.
Following [11], [17], a triple \((f,\lambda,\mu)\) of smooth functions on \(M\) is called a gradient generalized quasi-Einstein structure if \[R_{ij} + f_{ij} - \mu f_i f_j = \lambda g_{ij}, \label{eq:GQE2}\tag{3}\] where \(f_{ij} = \nabla_j\nabla_i f\) and \(f_i = \nabla_i f\). When \(\mu\) is the constant \(1/m\) for a positive real number \(m\), we speak of a gradient generalized \(m\)-quasi-Einstein manifold.
From 3 we immediately obtain the trace \[\Delta f = n\lambda - R + \mu|\nabla f|^2. \label{eq:trace}\tag{4}\] Contracting 3 with \(f^j\) yields a useful gradient formula: \[\frac{1}{2}\nabla_i|\nabla f|^2 = \lambda f_i - R_{ij}f^j + \mu|\nabla f|^2 f_i. \label{eq:grad}\tag{5}\]
We recall the two auxiliary functions introduced in [17] that govern the Laplacian of the scalar curvature. For a gradient GQE manifold we define \[\begin{align} I :=&\; (n-1)\Delta\lambda - 2\mu(n-1)\langle\operatorname{grad}f,\operatorname{grad}\lambda\rangle + \frac{1+2\mu}{2}\langle\operatorname{grad}R,\operatorname{grad}f\rangle \\ &\quad - (1-\mu)|\hat{\operatorname{Ric}}|^2 + \frac{(n\lambda-R)\bigl(R(1-\mu+\mu n) + \mu\lambda n(1-n)\bigr)}{n}, \end{align}\] where \(\hat{\operatorname{Ric}} = \operatorname{Ric}- \frac{R}{n}g\) is the trace-free Ricci tensor, and a similar expression for \(J\) (which we do not need explicitly because it will vanish under our assumptions). The next identity was proved in [17] (see also [18]).
Lemma 1. On any gradient GQE manifold, \[\frac{1}{2}\Delta R = I + J. \label{eq:DeltaR}\tag{6}\]
In the present paper we will restrict to the case where both the scalar curvature \(R\) and the coefficient \(\mu\) are constant. Under these assumptions all derivatives of \(R\) and \(\mu\) vanish; consequently \(J\equiv 0\) and \(\Delta R=0\), so Lemma 1 forces \(I=0\). For \(\mu=1/m\) this condition reads \[\begin{align} 0 = (n-1)\Delta\lambda &- \frac{2}{m}(n-1)\langle\operatorname{grad}f,\operatorname{grad}\lambda\rangle - \frac{m-1}{m}|\hat{\operatorname{Ric}}|^2 \\ &+ \frac{(n\lambda-R)\bigl(R(1+\frac{n-1}{m}) + \frac{\lambda n(1-n)}{m}\bigr)}{n}. \label{eq:Izero} \end{align}\tag{7}\] Equation 7 will be our starting point for the derivation of the Laplacian of \(v\).
We collect the classical results that will be used in the proofs. All manifolds are assumed to be complete and without boundary.
Lemma 2 (Yau [7]). Let \(u\) be a non-negative smooth subharmonic function on a complete Riemannian manifold \(M\). If \(u\in L^p(M)\) for some \(p>1\), then \(u\) is constant.
Lemma 3 (Schoen–Yau [8]). Let \(u\ge 0\) be a smooth subharmonic function on a complete manifold \(M\). For any ball \(B(q,2r)\subset M\) and any smooth cut-off function \(\phi\) supported in \(B(q,2r)\) with \(\phi\equiv 1\) on \(B(q,r)\) and \(|\nabla\phi|\le c/r\), there exists a universal constant \(A\) (depending only on \(c\)) such that \[\int_{B(q,r)}|\nabla u|^2 \le \frac{A}{r^2}\int_{B(q,2r)} u^2.\]
Lemma 4 (Caminha–Souza–Camargo [9]). Let \(X\) be a smooth vector field on a complete, non-compact, oriented Riemannian manifold \(M\). If \(\operatorname{div}X\) does not change sign on \(M\) and \(|X|\in L^1(M)\), then \(\operatorname{div}X\equiv 0\).
Remark 1. If \(M\) is not orientable, we may pass to the orientation double cover \(\widetilde{M}\). Every structure (metric, soliton functions, etc.) lifts isometrically, the lifted vector field \(\widetilde{X}\) satisfies the same hypotheses, and the conclusion \(\operatorname{div}\widetilde{X}=0\) on \(\widetilde{M}\) implies \(\operatorname{div}X=0\) on \(M\) by local isometry. Hence the lemma applies without loss of generality.
Lemma 5 (Karp [19]). Let \(u\ge 0\) be a smooth subharmonic function on a complete, non-compact Riemannian manifold with non-negative sectional curvature. If \(\int_M |\nabla u|^{\frac{n}{n-1}} < \infty\), then \(u\) is constant.
Lemma 6 (Kanai [20]). If a complete Riemannian manifold admits a non-homothetic conformal vector field, then it is isometric to Euclidean space.
Lemma 7 (Tashiro [21]). Let \((M,g)\) be a complete Riemannian manifold. If there exists a smooth function \(w\) on \(M\) such that \(\operatorname{Hess}w = c\,g\) for some constant \(c\neq 0\), then \(M\) is isometric to Euclidean space.
Finally, we need the existence of good cut-off functions under a mild Ricci lower bound. The following statement is a special case of results of Bianchi and Setti [22].
Lemma 8 ([22]). Assume that the Ricci curvature of a complete manifold \(M\) satisfies \[\operatorname{Ric}\ge -(n-1)\frac{k^2}{1+r^2}\, g\] for some constant \(k\), where \(r(x)=d(x,q)\) is the distance from a fixed point \(q\in M\). Then, for every large \(r>0\), there exists a smooth cut-off function \(\xi_r\) with compact support in \(B(q,2r)\) such that \[0\le \xi_r\le 1,\quad \xi_r\equiv 1 \text{ on } B(q,r),\quad |\nabla\xi_r|\le \frac{K}{r},\quad |\Delta\xi_r|\le \frac{K}{r^2},\] with a constant \(K>0\) independent of \(r\).
In this section we assume that \(M\) is a gradient generalized \(m\)-quasi-Einstein manifold with constant scalar curvature \(R\) and \(\mu=1/m\), and we set \[v = e^{-f/m}\lambda.\]
Lemma 9. Under the above assumptions, \[e^{f/m}\Delta v = \frac{1}{n-1}\left(\frac{m-1}{m}|\hat{\operatorname{Ric}}|^2 - \frac{R(n\lambda-R)}{n}\Bigl(1+\frac{n-1}{m}\Bigr)\right). \label{eq:key}\tag{8}\]
Proof. From 7 we solve for \(\Delta\lambda\): \[\begin{align} \Delta\lambda &= \frac{2}{m}\langle\operatorname{grad}f,\operatorname{grad}\lambda\rangle + \frac{m-1}{m(n-1)}|\hat{\operatorname{Ric}}|^2 \notag\\ &\quad - \frac{(n\lambda-R)\bigl(R(1+\frac{n-1}{m}) + \frac{\lambda n(1-n)}{m}\bigr)}{n(n-1)}. \label{eq:Deltalambda} \end{align}\tag{9}\] On the other hand, a direct computation using \(\operatorname{div}(\phi X)=\phi\operatorname{div}X+\langle\operatorname{grad}\phi,X\rangle\) gives \[\begin{align} e^{f/m}\Delta v &= e^{f/m}\operatorname{div}\bigl( e^{-f/m}(\nabla\lambda - \tfrac{\lambda}{m}\nabla f) \bigr)\\ &= \Delta\lambda - \frac{2}{m}\langle\operatorname{grad}f,\nabla\lambda\rangle - \frac{1}{m}\lambda\Delta f + \frac{1}{m^2}\lambda|\nabla f|^2. \end{align}\] Now use the trace identity 4 , \(\Delta f = n\lambda - R + \frac{1}{m}|\nabla f|^2\), to replace \(\Delta f\): \[e^{f/m}\Delta v = \Delta\lambda - \frac{2}{m}\langle\operatorname{grad}f,\nabla\lambda\rangle - \frac{n}{m}\lambda^2 + \frac{1}{m}\lambda R.\] Insert the expression for \(\Delta\lambda\) from 9 ; the gradient terms cancel and we obtain \[e^{f/m}\Delta v = \frac{m-1}{m(n-1)}|\hat{\operatorname{Ric}}|^2 - \frac{(n\lambda-R)\bigl(R(1+\frac{n-1}{m}) + \frac{\lambda n(1-n)}{m}\bigr)}{n(n-1)} - \frac{\lambda}{m}(n\lambda-R).\] The last two summands combine as follows. Let us compute \[\begin{align} &-\frac{(n\lambda-R)}{n(n-1)}\Bigl(R(1+\tfrac{n-1}{m}) + \frac{\lambda n(1-n)}{m}\Bigr) - \frac{\lambda}{m}(n\lambda-R)\\ &= -(n\lambda-R)\Bigl[ \frac{R(1+\frac{n-1}{m})}{n(n-1)} + \frac{\lambda n(1-n)}{m}\cdot\frac{1}{n(n-1)} + \frac{\lambda}{m} \Bigr]. \end{align}\] Since \(\frac{\lambda n(1-n)}{m}\cdot\frac{1}{n(n-1)} = \frac{\lambda(1-n)}{m(n-1)} = -\frac{\lambda}{m}\), the bracket simplifies to \[\frac{R(1+\frac{n-1}{m})}{n(n-1)} - \frac{\lambda}{m} + \frac{\lambda}{m} = \frac{R(1+\frac{n-1}{m})}{n(n-1)}.\] Thus the whole expression becomes \[-(n\lambda-R)\frac{R(1+\frac{n-1}{m})}{n(n-1)} = -\frac{R(n\lambda-R)}{n(n-1)}\Bigl(1+\frac{n-1}{m}\Bigr).\] Adding the first summand, we obtain exactly 8 . ◻
When \(R\le 0\), \(\lambda>0\) and \(m>1\), every term on the right-hand side of 8 is clearly non-negative; hence \(\Delta v\ge 0\). Since \(v=e^{-f/m}\lambda>0\), \(v\) is a positive subharmonic function.
Theorem 2. Let \((M^n,g)\) be a complete non-compact gradient generalized \(m\)-quasi-Einstein manifold with constant scalar curvature \(R\le 0\), \(\lambda>0\) and \(m>1\). If the weighted function \(v=e^{-f/m}\lambda\) belongs to \(L^p(M)\) for some \(p>1\), then \(M\) is isometric to \(\mathbb{R}^n\).
Proof. By Lemma 9, \(v>0\) is subharmonic. Yau’s lemma (Lemma 2) implies that \(v\) is constant; write \(v\equiv c>0\). Then \(\Delta v=0\), and the non-negative right-hand side of 8 must vanish term by term: \[\begin{gather} \frac{m-1}{m(n-1)}|\hat{\operatorname{Ric}}|^2 = 0,\qquad -\frac{R(n\lambda-R)}{n(n-1)}\Bigl(1+\frac{n-1}{m}\Bigr) = 0. \end{gather}\] Since \(m>1\), the first equality forces \(|\hat{\operatorname{Ric}}|^2=0\), i.e.\(\operatorname{Ric}= \frac{R}{n}g\). The second, together with \(n\lambda-R>0\) (because \(\lambda>0\) and \(R\le 0\)), gives \(R=0\). Hence \(\operatorname{Ric}= 0\).
The GQE equation reduces to \(\operatorname{Hess}f - \frac{1}{m} df\otimes df = \lambda g\), and \(v=c\) yields \(\lambda = c e^{f/m}\). Set \(w = m e^{-f/m}\). A direct computation gives \[\operatorname{Hess}w = \nabla( -\nabla f\, e^{-f/m}) = e^{-f/m}\bigl( -\operatorname{Hess}f + \frac{1}{m} df\otimes df \bigr) = -e^{-f/m}\lambda g = -c\,g.\] Thus \(w\) is a smooth function with \(\operatorname{Hess}w = -c\,g\), where \(c>0\). By Tashiro’s theorem (Lemma 7), \(M\) is isometric to Euclidean space. ◻
Theorem 3. Let \((M^n,g)\) be a complete non-compact gradient generalized \(m\)-quasi-Einstein manifold with constant scalar curvature \(R\le 0\), \(\lambda>0\) and \(m>1\). Assume that \(v\) is bounded above and that the volume of geodesic balls grows at most linearly, i.e.\(\operatorname{Vol}(B(q,r)) \le C r\) for some \(C>0\) and all \(r>0\). Then \(M\) is isometric to \(\mathbb{R}^n\).
Proof. Let \(v\le C_0\) on \(M\). By Lemma 3 applied to the subharmonic function \(v\), for any large \(r\) we have \[\int_{B(q,r)}|\nabla v|^2 \le \frac{16}{r^2}\int_{B(q,2r)} v^2 \le \frac{16 C_0^2}{r^2}\operatorname{Vol}(B(q,2r)).\] Using the linear volume growth, \(\operatorname{Vol}(B(q,2r))\le 2C r\), we obtain \[\int_{B(q,r)}|\nabla v|^2 \le \frac{32 C_0^2 C}{r},\] which tends to \(0\) as \(r\to\infty\). Thus \(\int_M |\nabla v|^2 = 0\), so \(\nabla v=0\) everywhere; consequently \(v\) is constant. The remainder of the proof is identical to that of Theorem 2. ◻
Theorem 4. Let \((M^n,g)\) be a complete non-compact gradient generalized \(m\)-quasi-Einstein manifold with constant scalar curvature \(R\le 0\), \(\lambda>0\) and \(m>1\). If \(|\nabla v|\in L^1(M)\), then \(M\) is isometric to \(\mathbb{R}^n\).
Proof. Apply Lemma 4 to the vector field \(X = \nabla v\) (after passing to the orientation cover if necessary, see Remark after Lemma 4). From Lemma 9 we have \(\operatorname{div}X = \Delta v \ge 0\), and by assumption \(|X| = |\nabla v|\in L^1(M)\). Thus \(\Delta v \equiv 0\). Vanishing of the non-negative right-hand side of 8 gives \(\hat{\operatorname{Ric}}=0\) and \(R=0\), so \(\operatorname{Ric}=0\).
The GQE equation becomes \(\operatorname{Hess}f - \frac{1}{m} df\otimes df = \lambda g\), and as shown in Section 5, for \(w=m e^{-f/m}\) we have \(\operatorname{Hess}w = -v\,g\). If \(v\) is constant, then \(\operatorname{Hess}w = -c\,g\) with \(c>0\) and Tashiro’s theorem (Lemma 7) applies. If \(v\) is not constant, then \(\nabla w\) is a non-homothetic conformal vector field (since \(\mathcal{L}_{\nabla w}g = 2\operatorname{Hess}w = -2v\,g\), and \(v\) is non-constant), and Kanai’s theorem (Lemma 6) forces the manifold to be Euclidean. In either situation \(M\) is isometric to \(\mathbb{R}^n\). ◻
Theorem 5. Let \((M^n,g)\) be a complete non-compact gradient generalized \(m\)-quasi-Einstein manifold with constant scalar curvature \(R\le 0\), \(\lambda>0\) and \(m>1\). Assume that the sectional curvature of \(M\) is non-negative and that \[\int_M |\nabla v|^{\frac{n}{n-1}} < \infty.\] Then \(M\) is isometric to \(\mathbb{R}^n\).
Proof. \(v\) is subharmonic and the sectional curvature is non-negative. By Karp’s lemma (Lemma 5), \(v\) must be constant. The conclusion follows exactly as in Section 5. ◻
Theorem 6. Let \((M^n,g)\) be a complete non-compact gradient generalized \(m\)-quasi-Einstein manifold with constant scalar curvature \(R\le 0\), \(\lambda>0\) and \(m>1\). Suppose that there exists a constant \(k\) such that \[\operatorname{Ric}\ge -(n-1)\frac{k^2}{1+r(x)^2}\,g,\] where \(r(x)=d(x,q)\) for a fixed point \(q\in M\), and that the function \(v\) satisfies the weighted integrability condition \[\int_{M\setminus B(q,r)} \frac{v(x)}{d(x,q)^2}\,dV_g(x) < \infty\] for all \(r>0\) sufficiently large. Then \(M\) is isometric to \(\mathbb{R}^n\).
Proof. Let \(\Phi = e^{f/m}\Delta v\); by Lemma 9, \(\Phi\ge 0\) and \(\Delta v = e^{-f/m}\Phi\). Thanks to the Ricci lower bound, Lemma 8 supplies cut-off functions \(\xi_r\) with the stated properties. Multiply the identity \(\Delta v = e^{-f/m}\Phi\) by \(\xi_r\) and integrate over \(M\): \[\int_M \xi_r e^{-f/m}\Phi \,dV_g = \int_M \xi_r \Delta v \,dV_g.\] Using Green’s second identity and the fact that \(\xi_r\) is compactly supported, this equals \(\int_M v \Delta \xi_r \,dV_g\). Thus \[0 \le \int_M \xi_r e^{-f/m}\Phi = \int_M v \Delta \xi_r = \int_{B(q,2r)\setminus B(q,r)} v \Delta \xi_r,\] since \(\xi_r\equiv 1\) on \(B(q,r)\) and \(\xi_r=0\) outside \(B(q,2r)\). Now \(|\Delta \xi_r|\le K/r^2\) and \(v>0\), so \[\int_{B(q,2r)\setminus B(q,r)} v \Delta \xi_r \le \frac{K}{r^2}\int_{B(q,2r)\setminus B(q,r)} v.\] In the annular region \(B(q,2r)\setminus B(q,r)\) we have \(d(x,q)\le 2r\), hence \(\frac{1}{r^2} = \frac{4}{(2r)^2} \le \frac{4}{d(x,q)^2}\). Consequently, \[\frac{K}{r^2}\int_{B(q,2r)\setminus B(q,r)} v \le 4K \int_{B(q,2r)\setminus B(q,r)} \frac{v}{d(x,q)^2}.\] By hypothesis, the right-hand side tends to \(0\) as \(r\to\infty\) (the integral over the complement of a ball goes to zero). Therefore \[\lim_{r\to\infty}\int_M \xi_r e^{-f/m}\Phi = 0.\] Because \(\xi_r\uparrow 1\) pointwise and the integrand is non-negative, the monotone convergence theorem yields \[\int_M e^{-f/m}\Phi = 0.\] Consequently \(\Phi\equiv 0\) almost everywhere, and by smoothness \(\Phi\equiv 0\) everywhere. Hence \(|\hat{\operatorname{Ric}}|^2=0\) and \(R=0\), so \(\operatorname{Ric}=0\). The rest of the proof is identical to that of Theorem 4. ◻
We present three examples. The first is a non‑trivial local solution of the GQE equations on a negatively curved warped product, with non‑constant \(v\), that explicitly verifies Lemma 9. The second confirms the algebraic identity on a flat local model where \(v\) is constant. The third provides a rigorous analysis of complete examples, showing that no complete flat example with \(\lambda>0\) exists and that the natural negatively curved warped products fail the positivity condition globally.
Example 1 (A non‑trivial local verification of Lemma 9). Let \(n=3\), \(m=5>1\), \(\gamma=1\), and consider the warped product metric \[g = dr^{2} + e^{2r}(dx^{2}+dy^{2})\] on the interval \(r\in(\ln(6/7),\,\ln 3)\). The scalar curvature of this metric is constant, \(R = -n(n-1)\gamma^{2} = -6\), and the fiber is flat (\(\kappa=0\)). Define \[f(r) = -5\ln(3-e^{r}),\qquad \lambda(r) = \frac{5e^{r}}{3-e^{r}} - 2.\] Because \(3-e^{r}>0\) on the chosen interval, \(f\) is smooth. Moreover \(\lambda(r)>0\) precisely when \(r>\ln(6/7)\) (since \(\lambda(r)=0\) iff \(5e^{r}=2(3-e^{r})\), i.e.\(7e^{r}=6\), \(r=\ln(6/7)\)). Thus on \((\ln(6/7),\ln 3)\) all the hypotheses \(R\le0\), \(\lambda>0\), \(m>1\) are satisfied.
A direct computation shows that \((f,\lambda)\) solves the GQE equation \(\operatorname{Ric}+ \operatorname{Hess}f - \frac{1}{5} df\otimes df = \lambda g\). Indeed, \(f' = y = \frac{5e^{r}}{3-e^{r}}\), and from the spherical component (with \(\kappa=0\), \(n=3\)) \[\frac{\psi'}{\psi} f' = \lambda + \frac{\psi''}{\psi} + (n-2)\frac{\psi'^{2} - \kappa}{\psi^{2}} \label{eq:spherical}\tag{10}\] we obtain \(\lambda = f' - 2\), which matches our definition. The radial component \[f'' - \frac{1}{5}(f')^{2} = \lambda + (n-1)\frac{\psi''}{\psi} \label{eq:radial}\tag{11}\] becomes \(f'' - \frac{1}{5}(f')^{2} = \lambda + 2\). Since \(f'' = \bigl(\frac{5e^{r}}{3-e^{r}}\bigr)' = \frac{15e^{r}}{(3-e^{r})^{2}}\), we have \[f'' - \frac{1}{5}(f')^{2} = \frac{15e^{r}}{(3-e^{r})^{2}} - \frac{25e^{2r}}{5(3-e^{r})^{2}} = \frac{15e^{r} - 5e^{2r}}{(3-e^{r})^{2}} = \frac{5e^{r}}{3-e^{r}} = \lambda+2,\] confirming the GQE system.
Now set \(v = e^{-f/5}\lambda\). One finds \[v = e^{\ln(3-e^{r})}\Bigl(\frac{5e^{r}}{3-e^{r}} - 2\Bigr) = (3-e^{r})\frac{5e^{r} - 2(3-e^{r})}{3-e^{r}} = 5e^{r} - 2(3-e^{r}) = 7e^{r} - 6,\] which is positive and non‑constant on \((\ln(6/7),\ln 3)\). The Laplacian of a radial function on this warped product is \(\Delta v = v'' + 2\frac{\psi'}{\psi}v'\), where \(\psi=e^{r}\). Hence \(v' = v'' = 7e^{r}\), and \[\Delta v = 7e^{r} + 2\cdot 1 \cdot 7e^{r} = 21e^{r}.\] Now compute the left‑hand side of 8 : \[e^{f/5} = \frac{1}{3-e^{r}},\qquad e^{f/5}\Delta v = \frac{21e^{r}}{3-e^{r}}.\] For the right‑hand side, \(\hat{\operatorname{Ric}}=0\) (the metric is Einstein) and \(R=-6\). Moreover \(n\lambda - R = 3\lambda + 6 = \frac{15e^{r}}{3-e^{r}}\). Thus \[\begin{align} \frac{1}{n-1}\Bigl(\frac{m-1}{m}|\hat{\operatorname{Ric}}|^{2} - \frac{R(n\lambda-R)}{n}\bigl(1+\tfrac{n-1}{m}\bigr)\Bigr) &= \frac{1}{2}\Bigl(0 - \frac{(-6)\cdot\frac{15e^{r}}{3-e^{r}}}{3}\bigl(1+\tfrac{2}{5}\bigr)\Bigr)\\ &= \frac{1}{2}\Bigl( \frac{6\cdot 15e^{r}}{3(3-e^{r})}\cdot \frac{7}{5} \Bigr)\\ &= \frac{1}{2}\cdot \frac{42e^{r}}{3-e^{r}} = \frac{21e^{r}}{3-e^{r}}, \end{align}\] which matches the left‑hand side. This explicit computation verifies the algebraic identity 8 in a situation where \(v\) is not constant and \(R<0\).
We note that this example cannot be extended to a complete manifold with \(\lambda>0\) everywhere, because \(\lambda(r)\) becomes negative for \(r<\ln(6/7)\) and the solution blows up at \(r=\ln 3\). This illustrates that while Lemma 9 holds locally in great generality, global solutions with the required positivity may be highly restricted — a theme explored in the next examples.
Example 2 (Local model on a Euclidean ball). Let \(n\ge 3\), \(m>1\) and \(\tau>0\). On flat \(\mathbb{R}^{n}\), consider the ball \(\Omega = \{x\in\mathbb{R}^{n}:|x|^{2} < 2m\tau\}\). Define \[f(x) = -m\ln\!\Bigl(\tau - \frac{|x|^{2}}{2m}\Bigr),\qquad \lambda(x) = \frac{1}{\tau - \frac{|x|^{2}}{2m}}.\] Write \(r^{2}=|x|^{2}\) and \(\rho = \tau - r^{2}/(2m)\). A direct computation gives \[f_{i} = \frac{x_{i}}{\rho},\qquad f_{ij} = \frac{\delta_{ij}}{\rho} + \frac{x_{i}x_{j}}{m\rho^{2}}.\] Hence \[\operatorname{Hess}f - \frac{1}{m} df\otimes df = \Bigl(\frac{1}{\rho} + \frac{r^{2}}{m\rho^{2}}\Bigr)g_{ij} - \frac{r^{2}}{m\rho^{2}}g_{ij} = \frac{1}{\rho}\,g_{ij} = \lambda g_{ij}.\] The metric is flat, so \(\operatorname{Ric}=0\) and \(R=0\). One easily checks \(v = e^{-f/m}\lambda = 1\); thus \(\nabla v=0\) and \(\Delta v=0\). The right‑hand side of 8 becomes \(\frac{1}{n-1}(0-0)=0\), confirming the algebraic identity. This example, although defined only on a ball, illustrates the precise cancellation of the gradient terms and shows the consistency of the local geometry when \(v\) is constant.
Example 3 (Complete examples: non‑existence on flat space, failure in warped models). We investigate whether complete, non‑compact gradient generalized \(m\)‑quasi‑Einstein manifolds with constant \(R\le 0\), \(\lambda>0\) and \(m>1\) can exist, beyond the Euclidean space itself.
Case 1: Flat \(\mathbb{R}^{n}\). Let \((\mathbb{R}^{n}, g_{\text{std}})\) be the Euclidean space. Assume there exist smooth functions \(f,\lambda\) with \(\lambda>0\) satisfying \[\operatorname{Hess}f - \frac{1}{m} df\otimes df = \lambda\,g. \label{eq:flat}\tag{12}\] Set \(\phi = e^{-f/m} > 0\). A direct calculation using 12 gives \[\operatorname{Hess}\phi = -\frac{1}{m}e^{-f/m}\bigl(\operatorname{Hess}f - \frac{1}{m} df\otimes df\bigr) = -\frac{v}{m}\,g, \label{eq:Hessphi}\tag{13}\] where \(v = e^{-f/m}\lambda > 0\). Thus \(\operatorname{Hess}\phi = -\frac{v}{m}g\) is negative definite everywhere, so \(\phi\) is a smooth concave function on \(\mathbb{R}^n\). Consequently \(-\phi\) is a convex function bounded above by \(0\) (since \(\phi>0\)). A classical Liouville-type theorem (see e.g.[4]) forces \(-\phi\) to be constant; hence \(\phi\) is constant. From 13 we then obtain \(v = -m\,\phi^{-1}\Delta\phi = 0\), which contradicts \(\lambda>0\). Therefore, no complete flat gradient \(m\)‑quasi‑Einstein manifold with \(\lambda>0\) exists (apart from the trivial vacuum case \(\lambda=0\) which is excluded by hypothesis).
Case 2: Warped products with constant negative scalar curvature. A natural non‑Euclidean setting is a complete manifold with \(R<0\). Consider the warped product metric on \(M=\mathbb{R}^{n}\) \[g = dr^{2} + \psi(r)^{2} g_{N},\] where \((N,g_{N})\) is an \((n-1)\)-dimensional Einstein manifold with \(\operatorname{Ric}_{N} = (n-2)\kappa\,g_{N}\), \(\kappa\in\{-1,0,1\}\), and \(\psi(r)>0\) is smooth on \([0,\infty)\) with \(\psi(0)=0\), \(\psi'(0)=1\) (to ensure smoothness at the origin). The standard complete metrics of constant negative scalar curvature \(R = -n(n-1)\gamma^{2}\) (\(\gamma>0\)) are obtained by: \[\psi(r) = \frac{\sinh(\gamma r)}{\gamma}\quad (\kappa=1),\qquad \psi(r) = e^{\gamma r}\quad (\kappa=0),\qquad \psi(r) = \frac{\cosh(\gamma r)}{\gamma}\quad (\kappa=-1).\] We search for radial functions \(f = f(r)\), \(\lambda = \lambda(r)\) satisfying the GQE equation \[\operatorname{Ric}+ \operatorname{Hess}f - \frac{1}{m} df\otimes df = \lambda g.\] The Ricci tensor of \(g\) is given by \[\operatorname{Ric}(\partial_{r},\partial_{r}) = -(n-1)\frac{\psi''}{\psi},\qquad \operatorname{Ric}(X,X) = \Bigl( -\frac{\psi''}{\psi} - (n-2)\frac{\psi'^{2} - \kappa}{\psi^{2}} \Bigr) g(X,X)\] for \(X\perp\partial_{r}\). The Hessian of a radial function is \[\operatorname{Hess}f = f''\,dr^{2} + \psi\psi' f' \,g_{N} = f''\,dr^{2} + \frac{\psi'}{\psi} f'\, (g - dr^{2}).\] Decomposing the GQE equation into radial and spherical components yields \[\begin{align} f'' - \frac{1}{m} (f')^{2} &= \lambda + (n-1)\frac{\psi''}{\psi}, \tag{14}\\ \frac{\psi'}{\psi} f' &= \lambda + \frac{\psi''}{\psi} + (n-2)\frac{\psi'^{2} - \kappa}{\psi^{2}}. \tag{15} \end{align}\] Eliminating \(\lambda\) gives a single ODE for \(y = f'\): \[y' - \frac{1}{m} y^{2} - \frac{\psi'}{\psi} y = (n-2)\Bigl( \frac{\psi''}{\psi} - \frac{\psi'^{2} - \kappa}{\psi^{2}} \Bigr). \label{eq:ODE}\tag{16}\] For each of the three standard \(\psi\)’s one checks directly that \[\frac{\psi''}{\psi} = \gamma^{2},\qquad \frac{\psi'^{2} - \kappa}{\psi^{2}} = \gamma^{2},\] so the right‑hand side of 16 vanishes identically. Thus the ODE reduces to \[y' - \frac{1}{m} y^{2} - \frac{\psi'}{\psi} y = 0.\] This Bernoulli equation is solved by setting \(u = y^{-1}\), giving \[u' + \frac{\psi'}{\psi} u = -\frac{1}{m},\] with integrating factor \(\psi\). Hence \[(\psi u)' = -\frac{\psi}{m},\qquad \psi u = -\frac{1}{m} \int \psi\,dr + C.\] Thus \[y = \frac{\psi}{C - \frac{1}{m} \int \psi\,dr}.\]
We now examine \(\lambda\) given by 15 : \[\lambda = \frac{\psi'}{\psi} y - \gamma^{2} - (n-2)\gamma^{2} = \frac{\psi'}{\psi} y - (n-1)\gamma^{2}.\] Now we evaluate each case separately.
\(\kappa=1\), \(\psi = \frac{\sinh(\gamma r)}{\gamma}\). Then \(\int \psi\,dr = \frac{1}{\gamma^{2}}\cosh(\gamma r)\), \(\frac{\psi'}{\psi} = \gamma\coth(\gamma r)\). Set \(D = m\gamma^{2} C\). Then \[C - \frac{1}{m\gamma^{2}}\cosh(\gamma r) = \frac{1}{m\gamma^{2}}(D - \cosh(\gamma r)),\] hence \[y = \frac{\frac{\sinh(\gamma r)}{\gamma}}{\frac{1}{m\gamma^{2}}(D - \cosh(\gamma r))} = \frac{m\gamma \sinh(\gamma r)}{D - \cosh(\gamma r)}.\] Then \[\frac{\psi'}{\psi} y = \gamma\coth(\gamma r) \cdot \frac{m\gamma \sinh(\gamma r)}{D - \cosh(\gamma r)} = \frac{m\gamma^{2} \cosh(\gamma r)}{D - \cosh(\gamma r)}.\] Thus \[\lambda(r) = \frac{m\gamma^{2} \cosh(\gamma r)}{D - \cosh(\gamma r)} - (n-1)\gamma^{2}.\] As \(r\to\infty\), \(\cosh\to\infty\), so \(D - \cosh \sim -\cosh\), and the fraction tends to \(-m\gamma^{2}\), giving \(\lambda \to -(m+n-1)\gamma^{2} < 0\). Hence \(\lambda\) becomes negative for large \(r\), violating \(\lambda>0\) on the whole manifold.
\(\kappa=0\), \(\psi = e^{\gamma r}\). Then \(\int \psi\,dr = \frac{1}{\gamma}e^{\gamma r}\), \(\frac{\psi'}{\psi} = \gamma\). Set \(D = m\gamma C\), then \(C - \frac{1}{m\gamma}e^{\gamma r} = \frac{1}{m\gamma}(D - e^{\gamma r})\), so \[y = \frac{e^{\gamma r}}{\frac{1}{m\gamma}(D - e^{\gamma r})} = \frac{m\gamma e^{\gamma r}}{D - e^{\gamma r}}.\] Therefore, \[\lambda = \gamma y - (n-1)\gamma^{2} = \frac{m\gamma^{2} e^{\gamma r}}{D - e^{\gamma r}} - (n-1)\gamma^{2}.\] As \(r\to\infty\), \(D - e^{\gamma r}\sim -e^{\gamma r}\), so the fraction tends to \(-m\gamma^{2}\), giving \(\lambda\to -(m+n-1)\gamma^{2}<0\). As \(r\to -\infty\), \(e^{\gamma r}\to 0\), so the fraction tends to \(0\), yielding \(\lambda\to -(n-1)\gamma^{2}<0\). Thus \(\lambda(r)\) is negative for all \(r\), and certainly never positive everywhere.
\(\kappa=-1\), \(\psi = \frac{\cosh(\gamma r)}{\gamma}\). Then \(\int \psi\,dr = \frac{1}{\gamma^{2}}\sinh(\gamma r)\), \(\frac{\psi'}{\psi} = \gamma\tanh(\gamma r)\). Set \(D = m\gamma^{2} C\), then \(C - \frac{1}{m\gamma^{2}}\sinh(\gamma r) = \frac{1}{m\gamma^{2}}(D - \sinh(\gamma r))\), giving \[y = \frac{\frac{\cosh(\gamma r)}{\gamma}}{\frac{1}{m\gamma^{2}}(D - \sinh(\gamma r))} = \frac{m\gamma \cosh(\gamma r)}{D - \sinh(\gamma r)}.\] Thus \[\lambda = \gamma\tanh(\gamma r)\cdot \frac{m\gamma \cosh(\gamma r)}{D - \sinh(\gamma r)} - (n-1)\gamma^{2} = \frac{m\gamma^{2} \sinh(\gamma r)}{D - \sinh(\gamma r)} - (n-1)\gamma^{2}.\] As \(r\to\infty\), \(\sinh(\gamma r)\to\infty\), so \(D - \sinh \sim -\sinh\), the fraction tends to \(-m\gamma^{2}\), hence \(\lambda\to -(m+n-1)\gamma^{2}<0\). At \(r=0\), \(\sinh(0)=0\), giving \(\lambda(0) = 0 - (n-1)\gamma^{2} <0\). Hence \(\lambda\) is negative everywhere.
In all three hyperbolic models, \(\lambda\) is either negative for all \(r\) or becomes negative at infinity, and in no case is \(\lambda>0\) globally. Thus none of these complete non‑Euclidean spaces provides an example satisfying the hypotheses of our rigidity theorems.
Conclusion. The above analysis shows that complete flat examples with \(\lambda>0\) do not exist, and the natural negatively curved warped products fail the positivity condition on \(\lambda\). Together with the local example 1, this indicates that while Lemma 9 is a universal algebraic fact, the global requirements \(\lambda>0\) and completeness are highly restrictive. The rigidity results proved in this paper are therefore far from vacuous; rather, they imply that any hypothetical complete non‑compact gradient \(m\)‑quasi‑Einstein manifold with \(R\le 0\), \(\lambda>0\), \(m>1\) that is not isometric to \(\mathbb{R}^{n}\) must be highly exotic, necessarily violating all the additional integrability or curvature conditions stated in Theorems 2–6. Whether such a manifold can be constructed remains an interesting open problem.