Souplet-Zhang and Hamilton type gradient estimates for nonlinear elliptic equations on smooth metric measure spaces

In this article we present new gradient estimates for positive solutions to a class of nonlinear elliptic equations involving the f-Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet-Zhang and Hamilton types respectively and are established under natural lower bounds on the generalised Bakry-\'Emery Ricci curvature tensor. From these estimates we derive amongst other things Harnack inequalities and general global constancy and Liouville-type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.


Introduction
In this paper we are concerned with deriving gradient estimates for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces and exploiting some of their local and global implications. In more specific terms, we aim to formulate and prove gradient estimates of Souplet-Zhang and Hamilton types for positive smooth solutions u to the nonlinear elliptic equation: on a smooth metric measure space (M, g, dµ), where M is a Riemannian manifold with dimension n ≥ 2, dµ = e −f dv g is a weighted measure, f is a smooth potential on M, g is the Riemannian metric and dv g is the usual Riemannain volume measure. Here ∆ f is the so-called f -Laplacian that acts on functions u ∈ C 2 (M) by ∆ f u = ∆u − ∇f, ∇u , and Σ = Σ(x, u) is a sufficiently smooth but otherwise arbitrary nonlinearity depending on the spatial variable x and the independent variable u.
The f -Laplacian appearing in (1.1) is a symmetric diffusion operator with respect to the invariant weighted measure dµ = e −f dv g . It arises naturally in a variety of contexts ranging from geometry, probability theory and stochastic processes to quantum field theory, statistical mechanics and kinetic theory [4,28,44,45,49]. It is a generalisation of the Laplace-Beltrami operator to the smooth metric measure space context and one recovers the latter exactly when the potential f is a constant.
Alternatively and more directly one can view the f -Laplacian as the infinitesimal generator of the Dirichlet form Here Ito's SDE theory or Dirichlet form theory imply the existence of a minimal diffusion process (X t : t < T ) on M with infinitesimal generator L = ∆ f = e f div(e −f ∇) and lifetime T . This diffusion process is the solution to the Ito-Langevin SDE where W t denotes the standard Riemannian Brownian motion on M and so as a result the f -heat equation ∂ t u = ∆u − ∇f, ∇u is then the backward Kolmogorov equation associated with (X t : t < T ). Note in particular that the f -heat semigroup e t∆ f here can be represented in terms of the expectation (1.4) and the equilibrium solutions to the f -heat equation ∂ t u = ∆u− ∇f, ∇u are precisely the f -harmonic functions ∆ f u = 0, the linear version of (1.1). Apart from the connection alluded to above there is also a close connection between symmetric diffusion operators and Schrödinger operators [28,49]. Indeed the operator L = ∆ f viewed in L 2 (M, dµ) is unitarily equivalent to the Schrödinger operator H = ∆−V with V = |∇f | 2 /4−∆f /2 viewed in L 2 (M, dv g ) through the unitary isomorphism L given by where by the Feynman-Kac formula and upon recalling H = ∆−V with V = |∇f | 2 /4− ∆f /2 from above we have for all v ∈ C ∞ c (M): This results in a correspondence between the L p -uniqueness of f -heat semigroup e t∆ f and the Schrödinger semigroup e tH which is fundamental in the well-posedness of their respective Cauchy problems. There are other important and close connections between these semigroups and evolution equations with one another and, e.g., the Fokker-Planck equation (the forward Kolmogorov equation associated with the diffusion process (X t : t < T ) above) that we do not discuss here. (See [4,28,19,44,45,49] and the references therein for further detail and discussion in this direction.) Returning to the smooth metric measure space context and attempting to describe the geometry of the f -Laplacian and its associated diffusion process, we next introduce the generalised Ricci curvature tensor for the triple (M, g, dµ) by setting Ric f (g) = Ric(g) + Hess(f ). (1.9) Here Ric(g) denotes the usual Riemannain Ricci curvature tensor of g [see (1.22)] and Hess(f ) = ∇∇f stands for the Hessian of f . The weighted Bochner-Weitzenböck formula relating the quantities ∆ f |∇u| 2 , ∆ f u and Ric f (∇u, ∇u) for any u ∈ C 3 (M) then asserts that with the norm of the Hessian on the right being the Hilbert-Schmidt norm of 2-tensors. As a result of (1.10) a curvature lower bound Ric f (g) ≥ kg together with the Cauchy-Schwarz inequality result in the curvature-dimension condition CD(k, ∞), that is, 1 The main objective in this paper is to develop local and global gradient estimates of Souplet-Zhang and Hamilton types along with Harnack inequalities for positive smooth solutions to (1.1). It is well-known that such estimates and inequalities form the basis for deriving various qualitative properties of solutions and are thus of great significance [18,25,27,35,52,54]. Such properties may include local and global bounds, eigenvalues and spectral asymptotics, Liouville-type results, heat kernel bounds and asymptotics, applications to the time evolution equations and/or geometric flows, characterisation of ancient and eternal solutions, analysis of singularities and much more (see [1,5,6,7,11,14,16,18,21,22,27,28,32,36,37,38,39,40,42,43,45,47,48] and the references therein for further details and discussion).
Whilst in formulating and proving gradient estimates one often works with a specific choice or type of nonlinearity with an explicit structure of singularity, regularity, growth and decay, in this article we keep the analysis and discussion on a fairly general level without confining to specific examples or choices. This way we firstly provide a unified treatment of the estimates and secondly examine more transparently how the structure and form of the nonlinearity ultimately influences and interacts with the estimates and the subsequent results. As such the approach and analysis undertaken here largely unify, extend and improve various existing results in the literature for specific choices of nonlinearities. We discuss this further in the overview below.
Gradient estimates for positive solutions to (1.1) with logarithmic type nonlinearities in the form: 12) along with their parabolic counterparts have been the subject of extensive studies (see, e.g., [30,34,50] and the references therein). The interest in such problems originate partly from its natural links with gradient Ricci solitons and partly from geometric and functional inequalities, most notably, the logarithmic Sobolev inequalities [4,19,44,54].
Recall that a Riemannian manifold (M, g) is said to be a gradient Ricci soliton iff there there exists a smooth function f on M and a constant µ ∈ R such that (cf. [9,13,29]) Ric f (g) = Ric(g) + Hess(f ) = µg. (1.13) The notion is clearly an extension of an Einstein manifold and has a fundamental role in the analysis of singularities of the Ricci flow [21,54]. Taking trace from both sides of (1.13) and using the contracted Bianchi identity leads one to a simple form of (1.12) with constant coefficients: ∆u + 2µu log u = λu for suitable constant λ and u = e f (see [30] for details). Other types of equations generalising and strengthening (1.12) are with real exponents r, p, q and have been studied in detail in [7,15,39,40,47]. Yamabe type equations ∆u+A(x)u p +B(x)u = 0 also come under the form (1.1) with a power-like nonlinearity. In [5] the equation ∆u + u p + Bu = 0 is studied on a compact manifold and under suitable conditions on Ric(g), n, p and B it is shown to admit only constant solutions. The equation ∆u + A(x)u p = 0 for 1 ≤ p < 2 ⋆ − 1 = (n + 2)/(n − 2) with n ≥ 3 is considered in [17] and it is proved that when Ric(g) ≥ 0 any non-negative solution to this equation must be zero. In [51] it is shown that the same equation with constant A > 0, p < 0 admits no positive solution when Ric(g) ≥ 0. For a detailed account on the Yamabe problem in geometry see [23,31]. The natural form of Yamabe equation in the setting of smooth metric measure spaces is For gradient estimates, Harnack inequalities and other counterparts of the above results we refer the reader to [10,47,53]. A far more general form of Yamabe equation is the Einstein-scalar field Lichnerowicz equation (see, e.g., [12,13]). Here when the underlying manifold has dimension n ≥ 3 this takes the form ∆u + A(x)u p + B(x)u q + C(x)u = 0 with p = (n + 2)/(n − 2) and q = (3n−2)/(n−2) while when n = 2 this takes the form ∆u+A(x)e 2u +B(x)e −2u +C(x) = 0. The Einstein-scalar field Lichnerowicz equation in the setting of smooth metric measure spaces can naturally be formulated as: and For gradient estimates, Harnack inequalities and Liouville-type results in this and related contexts we refer the reader to [14,39,40,48] and the references therein.
Plan of the paper. Let us bring this introduction to an end by briefly describing the plan of the paper. In the concluding part of Section 1 (below) we describe the notation and terminology as used in the paper. In Section 2 we present the main results of the paper, specifically, a local and global gradient estimate of Souplet-Zhang type for equation (1.1), followed by local and global Harnack inequalities, a local and global gradient estimate of Hamilton type for equation (1.1) and a general Liouville-type result with some concrete applications to particular classes of nonlinearities. The subsequent sections are then devoted to the detailed proofs respectively. Section 3 is devoted to the proof of Theorem 2.1 and Section 4 presents the proof of the local and global Harnack inequalities in Theorem 2.3. Section 5 is devoted to the proof of Theorem 2.4 and the Liouville-type result in Theorem 2.7, which is an application of the former, is proved in Section 6. Finally in Section 7 we present the proof of the global Hamilton bound in Theorem 2.11. We also discuss some applications and examples.
Notation. Fixing a reference point p ∈ M we denote by d = d p (x) the Riemannian distance between x and p and by r = r p (x) the geodesic radial variable with origin at p. We write B a (p) ⊂ M for the geodesic ball of radius a > 0 centred at p. When the choice of the point p is clear from the context we often abbreviate and write d(x), r(x) or B a respectively. For X ∈ R we write X + = max(X, 0) and X − = min(X, 0). Hence X = X + + X − with X + ≥ 0 and X − ≤ 0. For the sake of future reference and use in the gradient estimates to come, we also introduce the functional quantity where we have set that is, the maximum of the f -Laplacian of the radial variable r = r p (x), that is, ∆ f r, taken over the unit sphere centred at the reference point p.
We typically denote partial derivatives with subscripts, in particular, for the nonlinear function Σ = Σ(x, u) we make use of Σ x , Σ u , etc. and we reserve the notation Σ x for the function Σ(·, u) obtained by freezing the argument u and viewing it as a function of x only. In fact below we frequently speak of ∇Σ x , ∆Σ x and ∆ f Σ x .
For the sake of reader's convenience we recall that in local coordinates (x i ) we have the following formulae for the Laplace-Beltrami operator, Riemann and Ricci curvature tensors respectively: 20) and and We point out that here are the Christoffel symbols. Additionally g ij , |g| and g ij = (g −1 ) ij are the components, the determinant and the components of the inverse of the metric tensor g respectively. Note in particular that upon referring to (1.20) we also have for the f -Laplacian.

Statement of the main results
In this section we present the main results of the paper with some accompanying discussion. The detailed arguments and proofs are delegated to the subsequent sections. We begin with a local Souplet-Zhang type gradient estimates for (1.1) exploiting the role of the nonlinearity Σ in (1.1).
An important quantity appearing in this estimate [cf.
3) for the explicit formulations]. As a close inspection reveals, these terms directly link to the nonlinearity Σ and the solution u, and as will be seen later, they play a decisive role in global estimates, Harnack inequalities and the Liouville theorems that follow. Note that since u > 0 and B R ⊂ M is compact, u is bounded away from zero and bounded from above; hence, in particular, this quantity is finite. The other quantities appearing here are k and [γ ∆ f ] + relating to the curvature bound Ric f (g) ≥ −(n − 1)kg in the theorem, the potential f , and the geometry of the triple (M, g, dµ) [cf. (1.18)].
Then on B R/2 the solution u satisfies the estimate: Here C > 0 is a constant depending only on n and the quantities P Σ (u) and R Σ (u) are defined respectively by

2)
and The local estimate above has a global counterpart subject to the prescribed bounds in the theorem being global. The proof follows by passing to the limit R → ∞ in (2.1). The precise formulation of this is given in the following theorem.
Then u satisfies the following global estimate on M: |∇u| Here C > 0 depends only on n, the quantities R Σ (u) and P Σ (u) are as in (2.2) and (2.3) in Theorem 2.1 and the supremum in (2.4) is taken over M.
One of the useful consequences of the estimates above is the following elliptic Harnack inequality for bounded positive solutions to the nonlinear elliptic equation (1.1). Later on we will prove another version of this inequality using a different approach.
where the exponent γ ∈ (0, 1) in (2.5) is given explicitly by the exponential Additionally, subject to the prescribed global bounds in Theorem 2.2, for all x, y in M, we have the same inequality (2.5) now with the exponent In (2.6) and (2.7), d(x, y) is the geodesic distance between x and y and C > 0 is as in (2.1) and (2.4) respectively.
The next type of estimate we move on to is a local Hamilton-type gradient estimate for positive solutions to the nonlinear elliptic equation (1.1). Here the estimate makes use of two non-negative parameters α, β with appropriate ranges (see below for details). This gives the estimate flexibility and scope for later applications.
The important quantity appearing in the estimate (2.8) here is T 10) for the explicit description of the terms] which again directly links to the nonlinearity Σ, the parameters α, β and the solution u. The other quantities appearing here are k and [γ ∆ f ] + relating to the curvature bound Ric f (g) ≥ −(n − 1)kg in the theorem, the potential f , and the geometry of the triple (M, g, dµ).
Theorem 2.4. Let (M, g, e −f dv g ) be a complete smooth metric measure space satisfying Ric f (g) ≥ −(n − 1)kg in B R with k ≥ 0. Let u be a positive solution to the nonlinear elliptic equation (1.1). Then for every α ≥ 0, 0 < β < 1/(1 + α) the solution u satisfies on B R/2 the estimate: (2.8) Here C > 0 is a constant depending only on n, α, β and the quantities S Σ (u) and T Σ (u) are defined respectively by and Again, the local estimate above has a global counterpart, when the asserted bounds in the theorem are global. This is the content of the following theorem.
Theorem 2.5. Let (M, g, e −f dv g ) be a complete smooth metric measure space satisfying the global curvature lower bound Ric f (g) ≥ −(n − 1)kg on M with k ≥ 0. Assume u is a positive solution to the nonlinear elliptic equation (1.1). Then for every α ≥ 0 and 0 < β < 1/(1 + α), u satisfies the following global estimate on M: Here C > 0 depends only on n, α, β, the quantities S Σ (u) and T Σ (u) are as in (2.9) and (2.10) in Theorem 2.4 and the supremums in (2.11) are taken over M.
Remark 2.6. Before moving on to discussing some applications of the above results, let us pause briefly to make a comment on the form of the local estimates (2.1) and (2.8) under a different but closely related curvature condition. Towards this end, recall that the generalised Bakry-Émery m-Ricci curvature tensor associated with the f -Laplacian is defined for the range m ≥ n by the symmetric 2-tensor When m = n, to make sense of (2.12) one only allows constant functions f as admissible potentials, in which case ∆ f = ∆ and Ric n f (g) ≡ Ric(g), and when m = ∞, by formally taking the limit m → ∞ in (2.12) one sets Referring to (2.12) and (2.13) now, it is easily seen that a lower bound on Ric m f (g) is a stronger condition than a lower bound on Ric f (g) as the former implies the latter but not vice versa. It can be shown [see [ 14) can be removed from the right-hand sides respectively. This means that (upon adjusting the constant C > 0 which will now depend on m as well) the local estimates respectively take the the forms and As is readily seen this stronger curvature condition leads to a potentially faster decay for the R-dependent terms on the right-hand side of the estimates as R → ∞. Note also that for m = n where one recovers the Laplace-Beltrami operator with the Riemannian volume measure and the usual Ricci curvature lower bound Ric(g) ≥ −(n − 1)kg, our results and estimates for the nonlinear equation (1.1) are, to the best of our knowledge, new in this generality for Σ = Σ(x, u).
Let us now move on to discussing some applications of the above estimates. Here our focus will be on Liouville theorems and we start with a brief outline for such results in the linear case, namely, for f -harmonic functions under various curvature conditions, before moving to the full nonlinear case in (1.1).
To this end recall that a classical form of Liouville's theorem for harmonic functions asserts that any positive harmonic function on M = R n must be a constant. (Hence by linearity and translation also any harmonic function on R n that is bounded either from above or below.) The assertion is a straightforward consequence of the mean-value property: pick x = y in R n , R > 0 sufficiently large and r = R − |x − y|. Then, by the positivity of u and the inclusion B r (y) ⊂ B R (x) it is plain that Passing to the limit R ր ∞ (and noting R/r ց 1) gives u(y) ≤ u(x). Interchanging the roles of x and y gives u(x) ≤ u(y). Thus putting these together gives u(x) = u(y).
The arbitrariness of x, y now proves the assertion. Using gradient estimates the same Liouville property was proved by S.T. Yau in [52] for positive harmonic functions on a complete Riemennian manifold with non-negative Ricci curvature (see also Theorem 3.1 and Corollary 3.1 in [35]). An essentially similar type of gradient estimate can be used to prove that any positive f -harmonic function u (i.e., ∆ f u = 0) on a complete smooth metric measure space with Ric m f (g) ≥ 0 for some n ≤ m < ∞ must be a constant (see, e.g., [28]  12)] and that u is unbounded. It is proved in [6] that if u is a bounded f -harmonic function on M (e.g., a positive and bounded from above f -harmonic function) and Ric f (g) ≥ 0 then u must be a constant.
We can now formulate the following theorem which is the counterpart of the above for the full nonlinear equation (1.1). Later we will present and discuss some notable applications of the theorem for special classes of nonlinearities of particular interest. Note that since here the curvature bound is expressed as Ric f (g) ≥ 0, the positivity of u alone is not enough to grant the constancy of u as can be seen from the discussion on f -harmonic functions above [(1.1) with Σ = Σ(x, u) ≡ 0]. For analogous results under the stronger curvature bound Ric m f (g) ≥ 0 but without the boundedness assumption on u see [42].
The proof of Theorem 2.7 is a direct consequence of the gradient estimates established in Theorems 2.4 and 2.5 and is presented in Section 6. We end this section by giving some nice applications of the above theorem. Towards this end consider first a superposition of power-like nonlinearities with constant coefficients A j , B j and real exponents p j , q j for 1 ≤ j ≤ N in the form (2.17) A direct calculation for the quantity pertaining to T Σ (u) in Theorem 2.7 gives which is then easily seen to be non-negative, as required by Theorem 2.7, upon suitably restricting the ranges of A j , B j and p j , q j as formulated below. This conclusion improves and extends earlier results on Yamabe type problems (cf. [14,17,51,48]). Further applications and results in this direction will be discussed in a forthcoming paper (see also [39,40]).
Remark 2.9. Note that the upper and lower bounds on the exponents p j , q j are given by the same quantity 1 − β(α/2 + 1) which can be adjusted by optimising the parameters α, β within their respective range. In particular if R(u) ≡ 0 we only require A j ≥ 0 and p j < 1 (with α = 0, β ց 0) and if P(u) ≡ 0 we only require B j ≤ 0 and q j > 0 (with α = 0, β ր 1).
As another application consider a superposition of logarithmic type and power-like nonlinearities with real exponents p, q, s and constant coefficients A, B and γ ∈ C 1 (R) in the form Σ(u) = u s γ(log u) + Au p + Bu q . (2.20) A straightforward calculation for the quantity pertaining to T Σ (u) in Theorem 2.7 then gives and so a discussion similar to that given before along with an application of Theorem 2.7 leads to the following result. Assume that along the solution u we have the inequality γ ′ + [β(α/2 + 1) + s − 1]γ ≤ 0 along with A ≥ 0, B ≤ 0, p ≤ 1 − β(α/2 + 1) and q ≥ 1 − β(α/2 + 1) for some α ≥ 0 and 0 < β < 1/(1 + α). Then u must be a constant and Σ(u) = 0.
We end this section with the following global Hamilton-type result for equation (1.1) with a dimension free constant. With the aid of this bound one can then prove a global Harnack-type inequality. Note that here M is assumed to be closed. (2.23) Furthermore, there holds the following interpolation-type Harnack inequality

Proof of the Souplet-Zhang estimate in Theorem 2.1
Before proceeding to the proof of Theorem 2.1 we need some intermediate results. Lemma 3.1 and Lemma 3.2 below ultimately lead to an elliptic differential inequality for a suitable quantity built out of the solution u in Lemma 3.3 that plays a key role in the proof of the theorem.
Proof. This is an easy calculation and the proof is left to the reader.
Next referring to the first term in the expression on the last line in (3.3) we can write

As a result by substituting (3.4) back in (3.3) it follows that
Now as for the term ∆ f h appearing on the right in (3.5) by substitution using the equation and therefore a rearrangement of terms and basic considerations including the identity ∇Σ = Σ x + Ae h Σ u ∇h, leads to which is the desired conclusion. The proof is thus complete.
Proof. This is a straightforward consequence of the identity (3.2) in Lemma 3.2 and the curvature lower bound Ric f (g) ≥ −(n − 1)kg in the lemma.
Proof of Theorem 2.1. In order to prove Theorem 2.1, the elliptic differential inequality in Lemma 3.3 will be combined with a localisation and cut-off function argument along with an application of the maximum principle to a suitably defined localised function. Now pick a reference point p in M, fix R ≥ 2 and let r = r p (x) denote the geodesic radial variable with respect to p. Below we use the radially symmetric cut-off function φ(x) =φ(r(x)), (3.10) supported in B R = B R (p) ⊂ M. The functionφ =φ(r) appearing on the right-hand side of (3.10) is chosen to satisfy the following properties (i)φ ∈ C 2 ([0, ∞), R), suppφ ⊂ [0, R] and 0 ≤φ ≤ 1, (iii) for every 0 < ε < 1 there exists c ε > 0 such that the bounds, hold on [0, ∞). Our aim is now to establish the desired estimate at every x in B R/2 and for this we consider the localised function φH with φ as in (3.10) and H as in Lemma 3.2. To this end we start with ∆ f (φH) = φ∆ f H + 2 ∇H, ∇φ + H∆ f φ or after a slight adjustment, (3.12) An application of the inequality (3.9) in Lemma 3.3 and another adjustment of terms as above then gives Assume now that φH is maximised on B R at the point q. Since the function r = r(x) is only Lipschitz continuous at the cut locus of p, using a standard argument of Calabi (see, e.g., [35] p. 21), we can assume without loss of generality that q is not in the cut locus of p and hence φH is smooth at q for the application of the maximum principle. Moreover we assume (φH)(q) > 0 as otherwise the result is trivial in view of H(x) ≤ 0 in B R/2 . Now at the maximal point q we have ∆ f (φH) ≤ 0 and ∇(φH) = 0. From (3.13) it thus follows that , (3.14) at q or dividing through by 2(1 − h) ≥ 0 that The goal is now to use (3.15) to establish the required estimate at x. To this end we consider two separate cases. Firstly, the case d(q) ≤ 1 and next the case d(q) ≥ 1.
, for all x with d(x) ≤ R/2, where R ≥ 2 by property (ii)] all the terms involving derivatives of φ at q vanish (in particular ∇φ = 0 and ∆ f φ = 0). So as a result it follows from (3.15) that at the point q, we have the bound and so By an application of Young's inequality it then follows after rearranging terms that Hence by recalling H = |∇h| 2 /(1 − h) 2 and h = log(u/A), we arrive at the bound at x Evidently this a special case of the inequality (2.1) upon noting (2.2) and (2.3) and so the proof of the estimate is complete in this case.
Case 2. Upon referring to the right-hand side of (3.15), and noting the properties of φ as listed at the start we proceed onto bounding the full expression on the right-hand side on (3.15) in the case d(q) ≥ 1. Towards this end dealing with the first term first, we have In much the same way regarding the terms involving Σ upon setting P Σ (u) = |Σ x |/(Ae h ) we have firstly (3.18) and likewise for the subsequent terms, upon where in the last equation we have set Now for the term ∆ f φ we use the Wei-Wylie weighted Laplacian comparison theorem taking advantage of the fact that it only depends on the lower bound on Ric f (g) ( [46]). Indeed recalling 1 ≤ d(q) ≤ R, it follows from Ric f (g) ≥ −(n − 1)kg with k ≥ 0 and Theorem 3.1 in [46] that [see (1.19) for notation] whenever 1 ≤ r ≤ R [in particular at the point q]. Thus proceeding on to bounding −∆ f φ, upon referring to (3.10) and using (ii Next referring to the last term on the first line in (3.15), the above, along with 1−h ≥ 1, after an application of Young inequality gives Thus, by putting together the above fragments, namely, the bounds in (3.17)-(3.23), we obtain, after reverting to u = Ae h , the following upper bound on φH 2 at q, Finally, recalling that φH is maximised on B R at q, φ ≡ 1 on B R/2 and 0 ≤ φ ≤ 1 on B R we can write sup B R/2 (3.25) Hence upon noting H = |∇h| 2 /(1 − h) 2 and h = log u, the above result in Thus in either of the two cases above we have shown the estimate to be valid at x. The arbitrariness of x in B R/2 gives the required conclusion and completes the proof.
Remark 3.4. Under the stronger curvature assumption Ric m f (g) ≥ −(m − 1)kg with n ≤ m < ∞ and k ≥ 0, we have from the Wei-Wylie weighted Laplacian comparison theorem (see also Corollary 1.2 in [28] as well as [29,33]) the f -Laplacian inequality [Compare with (3.21).] Therefore, for the cut-off function φ in (3.10), by virtue of φ being radial andφ ′ ≤ 0, we can deduce the lower bound From the bound √ k coth( √ kr) ≤ √ k coth( √ kR/2) ≤ (2 + √ kR)/R for R/2 ≤ r ≤ R (here we are making use of v coth v ≤ 1 + v and the monotonicity of coth v for v > 0) upon notingφ ′ ≡ 0 for 0 ≤ r ≤ R/2 it follows that Thus where in the second inequality we have made use of R ≥ 2 [compare with (3.22)]. Making use of this inequality in (3.23) instead of the previously used (3.22) implies that in (3.24) and hence (3.26) we can remove the term (2.14) at the cost of adjusting the constant C > 0 (so that now it will depend on m too).

Proof of the Harnack inequality in Theorem 2.3
In this section as the title suggests we give a proof of the local and global Harnack inequalities in Theorem 2.3. In its local form this is a consequence of the local estimate in Theorem 2.1 and in its global form this is a consequence of the global estimate in Theorem 2.2.
Towards this end pick x, y in M and let ζ = ζ(s) with 0 ≤ s ≤ 1 be a shortest geodesic curve with respect to the metric g joining x, y in B R/2 ⊂ M. Thus ζ(0) = x, ζ(1) = y and ζ(s) ∈ B R/2 for all 0 ≤ s ≤ 1.

Proof of the Hamilton estimate in Theorem 2.4
Before moving on to the proof of Theorem 2.4 we pause to state and prove two useful lemmas that are needed later on in establishing an elliptic differential inequality for a suitable quantity built out of the solution u [see (5.7) in the proof of Theorem 2.4].
Lemma 5.2. Under the assumptions of Lemma 5.1 on u and h = u β with 0 < β < 1, the function G = G β α = h α |∇h| 2 with α ≥ 0 satisfies the equation Proof. In order to calculate the action of ∆ f on G we first write G = |∇h α | 2 wherē h α = 2h α/2+1 /(α + 2). Indeed here we have the relations ∇h α = h α/2 ∇h and Moreover a straightforward calculation gives the Hessian Now applying the weighted Bochner-Weitzenböck formula (1.10) toh α and recalling the relation G = |∇h α | 2 gives A rearrangement of terms now results in which is immediately seen to be the required conclusion.
Proof Theorem 2.4. From the curvature lower bound Ric f (g) ≥ −(n − 1)kg and the inequality h > 0, upon substituting into the identity (5.2) in Lemma 5.2 and making use of (5.1) in Lemma 5.1, it follows that Abbreviating hereafter the arguments of Σ and its partial derivatives for convenience we can proceed by writing where the last term on the right-hand side of (5.8) can in turn be calculated as Substituting the descriptions of the inner products (5.8) and (5.9) back in (5.7) then result in the inequality , and substituting accordingly in the first line on the righthand side of (5.10), we can rewrite the latter inequality as Let us denote by Z Σ the sum of the last three terms on the right-hand side of (5.11), and note upon recalling G = h α |∇h| 2 and u = h 1/β that Since ∇h, ∇G = ∇h, ∇(h α |∇h| 2 ) = αh α−1 |∇h| 4 + h α ∇h, ∇|∇h| 2 by substituting these back in the inequality (5.11) we can then write (5.13) Next, localising by taking a space-time cut-off function φ as in (3.10) and following a similar procedure to those used in the proof of Theorem 2.1, we can write Let q be a maximum point for φG in B R . For the sake of establishing the estimate at x in B R/2 we confine to the case d(q) ≥ 1 noting (φG)(q) > 0 . Now at the maximum point q we have the inequalities ∆ f (φG) ≤ 0 and ∇(φG) = 0. Therefore applying these to (5.14) and rearranging the inequality we have We now proceed onto bounding from above each of the terms on the right-hand side of (5.15). Again, the argument proceeds by considering two case d(q) ≤ 1 and d(q) ≥ 1, and so as noted above, in view of certain similarities with the proof of Theorem 2.1, we shall remain brief, focusing on case two only and mainly so on the differences. Towards this end, the first two terms on the right-hand side of (5.15) are seen to be bounded, directly in modulus, upon using the Cauchy-Schwarz and Young inequalities, by the expressions: where we have used √ G = h α/2 |∇h|, and in much the same way, Regarding the fourth term, i.e., the one involving Z Σ , we have upon substitution from (5.12) where in the last line we have written Lastly, for the term involving ∆ f φ, we proceed similar to the proof of Theorem 2.1, where by recalling the weighted Laplacian comparison theorem as in Section 3, we have Hence for the remaining term on the right-hand side of (5.15) upon recalling 0 ≤ φ ≤ 1 and adjusting the constant C > 0 if necessary we can write Having now estimated each of the individual terms on the right-hand side of (5.15) we proceed next by substituting these back into the inequality and finalising the estimate. Indeed, substituting (5.16)-(5.19) in (5.15), adding the expressions on the right-hand side of the former four inequalities and making note of the basic relation 20) it follows that at the point q we have Next, by the maximality of the localised function φG on B R at q, we have the chain of inequalities [recall that φ ≡ 1 on B R/2 and 0 ≤ φ ≤ 1 on B R ] sup B R/2 φG) 2 = (φG) 2 (q) ≤ (φG 2 )(q). (5.22) Hence combining the latter with (5.21) and making note of the relations h = u β and G = h α |∇h| 2 = β 2 u (α+2)β−2 |∇u| 2 and introducing s(α, β) = (1 + α)[1 − β(1 + α)]/β it follows that (Note that the condition 0 < β < 1/(1 + α) guarantees that (1 + α)[1 − β(1 + α)] > 0 and so in turn s(α, β) > 0.) This upon rearranging terms and taking square roots gives the desired estimate for every x ∈ B R/2 and so completes the proof.
Remark 5.3. A similar argument to the one outlined in Remark 3.4 implies that subject to the stronger curvature assumption there, in the local estimate above the functional term (2.14) can be removed from the right-hand side in (2.8). See also Remark 2.6.
Proof of Theorem 2.11. If u > 0 is constant then (2.23) is trivially true so we assume u is non-constant. Now since for any σ ≥ e we have u ≤ σA, an application of Lemma 7.1 above, upon replacing A with σA and noting Σ x ≡ 0 gives, Since M is closed an application of the maximum principle now gives P γ [u] ≤ 0 or else that P γ [u] is constant. However since u is non-constant and σ ≥ e is arbitrary, the function P γ [u] = γ|∇u| 2 /u − u[log(A/u) + log σ] is constant for at most one σ ≥ e. Thus we can write |∇ log u| 2 = |∇u| 2 /u 2 ≤ 2k[log(A/u) + log σ] (7.12) and so by passing to the limit σ ց e and noting that M is compact we obtain (2.23). Next to prove (2.24) set Z(x) = log[(Ae)/u] = 1 + log(A/u). Then a straightforward calculation and making use of (2.23) gives Integrating the above along a minimising geodesic joining a pair of points p, q in M then gives log(Ae/u(q)) ≤ log(Ae/u(p)) + d(p, q) k/2. (7.14) For any ε > 0 thus log(Ae/u(q)) ≤ (1 + ε)[log(Ae/u(p)) + d 2 (p, q)k/(2ε)]. Exponentiating and rearranging yields the desired inequality (2.24).
Acknowledgement. The authors gratefully acknowledge support from EPSRC. They also wish to thank the anonymous referee for a careful reading of the manuscript and useful comments.