Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

Ali Taheri

doi:10.1515/acv-2020-0099

Article Open Access

Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

Ali Taheri

Published/Copyright: September 3, 2021

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From the journal Advances in Calculus of Variations Volume 16 Issue 2

Abstract

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a ( k , m ) -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results.

Keywords: Smooth metric measure spaces; super Ricci flow; Witten Laplacian; gradient estimates; Liouville type results

MSC 2010: 53C44; 58J60; 58J35; 60J60

1 Introduction

In this paper, we establish elliptic type gradient estimates for positive solutions to a class of nonlinear parabolic equations involving the Witten Laplacian in the context of smooth metric measure spaces where both the metric and potential are time dependent. We also discuss some applications and consequences of these estimates, most notably to Liouville theorems and global estimates.

Gradient estimates have a central place in geometric analysis with a vast scope of applications (for some recent work, see [2, 6, 9, 11, 14, 17, 15, 5, 23, 27, 29, 31, 32, 3, 4, 10, 13, 26, 34] and the references therein). The estimates of interest in this paper fall under the category of Souplet–Zhang and Hamilton type estimates that were first formulated and proved for the heat equation on static manifolds in [11, 23]. Such estimates along with the differential Harnack and Li–Yau type inequalities have since been extensively studied and applied to families of parabolic equations on manifolds with both static and evolving metrics (e.g., under the Ricci flow and its relatives). They are well known not only for their effectiveness in proving classical Harnack inequalities, Liouville theorems and sharp bounds on heat kernels to mention a few, but also more recently and in conjunction with the Perelman entropy method for establishing sharp geometric and functional inequalities in a variety of contexts and settings (see, e.g., [4, 10, 13, 17, 15, 16, 18, 28, 34]). In this paper, we make a contribution to the subject by deriving such estimates in the context of smooth metric measure spaces equipped with time dependent metrics and potentials under suitable bounds on the generalized Bakry–Émery curvature tensor before presenting some applications, most notably here to Liouville theorems.

To this end, suppose that ( M , g , e - f ⁢ d ⁢ v ) is a smooth metric measure space, by which it is meant that ( M , g ) is a complete Riemannian manifold of dimension n (here n ≥ 2 ), f is a smooth potential on M and d ⁢ μ = e - f ⁢ d ⁢ v is a weighted measure on M conformally equivalent to the Riemannian volume measure d ⁢ v = d ⁢ v g . (For the significance and role of smooth metric measure spaces in modern geometric analysis see [3, 4, 10, 22, 28], the sizeable recent literature, e.g., [8, 15, 16, 19, 24, 30, 33, 34] as well as the discussion in the next section). A ( k , m ) -super Perelman–Ricci flow on M refers to the evolution of a one parameter family of smooth metric-potential pairs ( g , f ) on M subject to the relation

(1.1) 1 2 ⁢ ∂ ⁡ g ∂ ⁡ t ⁢ ( x , t ) + Ric f m ⁡ ( g ) ⁢ ( x , t ) ≥ - k ⁢ g ⁢ ( x , t ) , t > 0 .

Here k, m are real constants with m ≥ n (not necessarily an integer). The second-order symmetric tensor field Ric f m ⁡ ( g ) is the Bakry–Émery m-Ricci curvature tensor given in this evolutionary context by (cf. also (2.2) below)

(1.2) Ric f m ⁡ ( g ) ⁢ ( x , t ) = Ric ⁡ ( g ) ⁢ ( x , t ) + ∇ g ⁢ ( t ) ⁡ ∇ g ⁢ ( t ) ⁡ f ⁢ ( x , t ) - ∇ g ⁢ ( t ) ⁡ f ⊗ ∇ g ⁢ ( t ) ⁡ f m - n ⁢ ( x , t ) ,

with Ric ⁡ ( g ) the usual Ricci tensor and ∇ g ⁡ ∇ g ⁡ f the Hessian of f. Note that the definition in (1.2) extends to m = n only for constant functions f hence giving Ric f n ⁡ ( g ) ≡ Ric ⁡ ( g ) , and to m = ∞ by

Ric f ⁡ ( g ) = Ric f ∞ ⁡ ( g ) ≡ Ric ⁡ ( g ) + ∇ g ⁡ ∇ g ⁡ f .

In this paper, we consider positive smooth solutions to the semilinear parabolic equation on M:

(1.3) ∂ ⁡ u ∂ ⁡ t - Δ f ⁢ u = F ⁢ ( u ) , t > 0 ,

where ℒ = Δ f denotes the Witten Laplacian (also known as Nelson’s diffusion operator in stochastic mechanics, or else the weighted Laplacian). It acts on smooth functions v ∈ 𝒞 ∞ ⁢ ( M ) by ^[1]

(1.4) Δ f ⁢ v = e f ⁢ div ⁡ ( e - f ⁢ ∇ ⁡ v ) = Δ ⁢ v - 〈 ∇ ⁡ f , ∇ ⁡ v 〉 ,

and constitutes a natural extension of the Laplace–Beltrami operator to the context of smooth metric measure spaces. The Witten Laplacian has close links with Markov processes and probability theory, quantum field theory, Riemannian geometry and more modern areas in mathematical physics. Our goal here is to establish among other things local and global elliptic type gradient estimates for positive smooth solutions to (1.3) when both the metric and potential evolve under a ( k , m ) -super Perelman–Ricci flow (see Theorem 3.1) and discuss some implications of these estimates.

Regarding the nonlinearity in (1.3), we take F = F ⁢ ( u ) a sufficiently smooth function on the half-axis u > 0 . Of particular interest, prompted by the surge of recent research work in the literature, is when F has a power-like growth and/or a logarithmic singularity at infinity, e.g., F ⁢ ( u ) = A ⁢ u p ⁢ log ⁡ u + B ⁢ u q with real p , q , or F ⁢ ( u ) = A ⁢ u ⁢ | log ⁡ u | p + B ⁢ u q with real p , q and p > 1 (see Sections 5 and 6 for more details). It turns out that a crucial role in the estimates is played here by the non-negative, F and u dependent quantity,

(1.5) 𝖱 F ⁢ ( u ) = [ 2 ⁢ u ⁢ ( 1 + u ) ⁢ F ′ ⁢ ( u ) - ( 1 + 2 ⁢ u ) ⁢ F ⁢ ( u ) 2 ⁢ ( 1 + u ) 2 ] + .

We discuss some consequences of these estimates and other closely related issues. In particular, and as a byproduct, we establish some new relative evolutionary estimates that include a Hamilton–Zhang type global gradient estimate to the nonlinear context with dimension free constants. As the static case is a particular instance of the system, we also discuss consequences of the gradient estimate to that context. In particular, here we establish new Liouville type results for the stationary elliptic counterpart of (1.3): Δ f ⁢ u + F ⁢ ( u ) = 0 subject to the non-negativity of the Bakry–Émery curvature tensors Ric f m ≥ 0 in Section 5 and global curvature free gradient estimates for the full parabolic equation (1.3) in Section 6 (see in particular Theorem 6.1 and the subsequent discussion). The non-negativity and/or monotonicity of related entropy like quantities is a question of great contemporary interest [7, 1, 8, 17, 22, 26, 28, 34].

2 Preliminaries

Let ( M , g ) be a complete n-dimensional Riemannian manifold. Given f ∈ 𝒞 ∞ ⁢ ( M ) , put d ⁢ μ = e - f ⁢ d ⁢ v g where d ⁢ v g is the Riemannian volume measure. The triple ( M , g , d ⁢ μ ) is then called a smooth metric measure space, or equivalently a weighted manifold (see [3, 4, 10, 19] for nice and exquisite introductions to the subject).^[2]

The f-Laplacian (1.4) is the natural substitute for the Laplace–Beltrami operator in this context (in fact, as is easily seen, the two coincide when the potential f is constant). It is a symmetric diffusion operator with respect to the invariant measure d ⁢ μ = e - f ⁢ d ⁢ v g [borrowing the terminology from the theory of Dirichlet forms and Itô’s SDE theory prompted by the second integral in (2.1)], and as one readily verifies for u , v ∈ 𝒞 0 ∞ ⁢ ( M ) :

(2.1) ∫ M e - f ⁢ v ⁢ Δ f ⁢ u ⁢ 𝑑 v g = ∫ M - e - f ⁢ 〈 ∇ ⁡ u , ∇ ⁡ v 〉 ⁢ d ⁢ v g = ∫ M e - f ⁢ u ⁢ Δ f ⁢ v ⁢ 𝑑 v g .

Next regarding the curvature properties of ( M , g , d ⁢ μ ) , the theory of carré du champ for Markov diffusion operators and the curvature dimension condition CD ⁡ ( k , m ) as developed by Bakry–Émery (see [3, 4]) naturally lead to the generalized m-Ricci curvature tensor Ric f m = Ric f m ⁡ ( g ) associated with the f-Laplacian defined by

(2.2) Ric f m ⁡ ( g ) = Ric ⁡ ( g ) + ∇ ⁡ ∇ ⁡ f - ∇ ⁡ f ⊗ ∇ ⁡ f m - n .

When m = n , to make sense of (2.2) one only allows constant functions f as admissible potentials, in which case Δ f = Δ and Ric f n ⁡ ( g ) ≡ Ric ⁡ ( g ) . By contrast when m = ∞ , one simply defines

Ric f ⁡ ( g ) := Ric f ∞ ⁡ ( g ) = Ric ⁡ ( g ) + ∇ 2 ⁡ f .

These different Ricci type tensors carry important geometric information and are of great utility in the theory. Note that here we have at our disposal the weighted version of the Bochner–Weitzenböck formula

(2.3) 1 2 ⁢ Δ f ⁢ | ∇ ⁡ u | 2 = | ∇ ⁡ ∇ ⁡ u | 2 + 〈 ∇ ⁡ u , ∇ ⁡ Δ f ⁢ u 〉 + Ric f ⁡ ( ∇ ⁡ u , ∇ ⁡ u ) .

Upon switching the inequality sign in (1.1) to equality, we get the ( k , m ) Perelman–Ricci flow ((2.4) below). Under the additional assumption that the weighted measure d ⁢ μ = e - f ⁢ d ⁢ v remains static in time, it can be seen that

∂ ⁡ d ⁢ μ t ∂ ⁡ t = ∂ ∂ ⁡ t ⁢ ( e - f ⁢ ( t ) ⁢ d ⁢ v ) = [ - ∂ ⁡ f ∂ ⁡ t + 1 2 ⁢ Tr ⁡ ( ∂ ⁡ g ∂ ⁡ t ) ] ⁢ d ⁢ μ t = 0 .

Here Tr stands for the metric trace on ( 0 , 2 ) -tensors, and the combined system resulting from the flow and the stationary volume measure constraint takes the form

(2.4) 1 2 ⁢ ∂ ⁡ g ∂ ⁡ t ⁢ ( x , t ) + Ric f m ⁡ ( g ) ⁢ ( x , t ) = - k ⁢ g ⁢ ( x , t ) ,

(2.5) ∂ ⁡ f ∂ ⁡ t ⁢ ( x , t ) - 1 2 ⁢ Tr ⁡ ( ∂ ⁡ g ∂ ⁡ t ) ⁢ ( x , t ) = 0 .

In particular, it follows from (2.4) and (2.5) that the potential f satisfies the evolution equation

∂ ⁡ f ∂ ⁡ t + Δ ⁢ f = Tr ⁡ ( 1 2 ⁢ ∂ ⁡ g ∂ ⁡ t + ∇ ⁡ ∇ ⁡ f ) = - R + | ∇ ⁡ f | 2 m - n - n ⁢ k .

The case ( k , m ) = ( 0 , ∞ ) in system (2.4)–(2.5) is what was introduced by Perelman in [22] as the L 2 -gradient flow of the functional 𝒫 ⁢ ( g , f ) = ∫ M ( R + | ∇ ⁡ f | 2 ) ⁢ e - f ⁢ 𝑑 v g subject to the stationary volume measure constraint considered above. Here R = R ⁢ ( g ) is the scalar curvature, Ric f m ⁡ ( g ) = Ric ⁡ ( g ) + ∇ 2 ⁡ f is the modified Ricci tensor, and f is the potential satisfying the so-called conjugate or adjoint heat equation,

□ ⋆ ⁢ f = - ∂ ⁡ f ∂ ⁡ t - Δ ⁢ f + R = 0 .

The above discussion illustrates another aspect of how the problems considered here link with the larger and more recent body of research on the subject and the remarkable work on the Poincaré conjecture [22]. For further reading and related work with volume measure constraints, see [8, 12, 17, 15, 21, 27, 34] and the references therein.

Notation.

For any pair of points x , y on M we designate by d = d g ⁢ ( t ) ⁢ ( x , y ) or d ⁢ ( x , y ; t ) the Riemannian distance between x, y with respect to the metric g = g ⁢ ( t ) . Fixing a reference point x 0 on M, we denote by ϱ = ϱ ⁢ ( x , t ) the geodesic radial variable measuring the distance between x and x 0 at time t. For R , T > 0 we introduce the compact set

ℬ R , T ≡ ℬ R , T ⁢ ( x 0 ) ≡ { ( x , t ) : d ⁢ ( x , x 0 ; t ) ≤ R , 0 ≤ t ≤ T } ⊂ M × [ 0 , T ] .

Throughout, we assume R ≥ 2 and we make use of the notation s + = max ⁡ ( s , 0 ) and s - = max ⁡ ( - s , 0 ) .

3 A Hamilton–Souplet–Zhang type gradient estimate

The main result here is a local elliptic gradient estimate on positive smooth solutions to equation (1.3). Here the metric g = g ⁢ ( t ) and the potential f = f ⁢ ( t ) evolve under the ( k , m ) -super Perelman–Ricci flow (1.1) (see the discussion following the theorem), and to make it clear the fact that both the metric and the potential are time dependent means that in explicit terms Δ f = Δ g ⁢ ( t ) - 〈 ∇ g ⁢ ( t ) ⁡ f , ∇ g ⁢ ( t ) 〉 (cf. (1.4)). For the purpose of Theorem 3.1 below, x 0 ∈ M and R ≥ 2 are chosen and are fixed whilst T > 0 and ℬ R , T is as defined above. The estimate makes use of the constants 𝗁 , 𝗄 ≥ 0 describing the lower bound Ric f m ⁡ ( g ) ≥ - 𝗄 m ⁢ g with 𝗄 m = ( m - 1 ) ⁢ 𝗄 or ( n - 1 ) ⁢ 𝗄 depending on m < ∞ or m = ∞ , respectively, and ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ 𝗁 ⁢ g in the compact set ℬ R , T . Note finally that due to the local nature of the estimate here all one needs is for u to be a positive solution in an open set containing ℬ R , T .

Theorem 3.1.

Let u be a positive solution to (1.3) and let { ( M , g ⁢ ( t ) , f ⁢ ( t ) ) : t ∈ [ 0 , T ] } be a complete solution to the super Perelman–Ricci flow (1.1) with Ric f m ⁡ ( g ) ≥ - k m ⁢ g and ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ h ⁢ g for some h , k m ≥ 0 in B R , T . Then there exists C > 0 depending only on n, m such that, for all ( x , t ) ∈ B R / 2 , T with t > 0 ,

(3.1) | ∇ ⁡ u | u + 1 ≤ C ⁢ { ( 1 R + ζ R + 𝖪 + 1 t ) ⁢ ( sup ℬ R , T ⁡ u ) + sup ℬ R , T ⁡ 𝖱 F ⁢ ( u ) } .

Here

𝖱 F ⁢ ( u ) = { 2 ⁢ u ⁢ ( 1 + u ) ⁢ F ′ ⁢ ( u ) - ( 1 + 2 ⁢ u ) ⁢ F ⁢ ( u ) 2 ⁢ ( 1 + u ) 2 } + ,

𝖪 = 𝗁 2 + 𝗄 2 and ζ = [ Z Δ f ] + where

Z Δ f = max ( x , t ) ⁡ { Δ f ⁢ ϱ ⁢ ( x , t ) : d ⁢ ( x , x 0 ; t ) = 1 , 0 ≤ t ≤ T } .

Moreover, in the case m < ∞ one can remove the term ζ / R in (3.1).

Remark 3.2.

From the bounds Ric f m ⁡ ( g ) ≥ - 𝗄 m ⁢ g and ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ 𝗁 ⁢ g , one obtains (1.1) with k = 𝗄 m + 𝗁 , i.e., k = ( m - 1 ) ⁢ 𝗄 + 𝗁 for m < ∞ , and k = ( n - 1 ) ⁢ 𝗄 + 𝗁 for m = ∞ . The lower bound Ric f m ⁡ ( g ) ≥ - 𝗄 m ⁢ g here is required for the Wei–Wylie weighted Laplacian comparison theorem (step (v) in the proof) and the bound ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ 𝗁 ⁢ g for controlling the time derivative of the geodesic distance, namely d t = ∂ ⁡ [ d g ⁢ ( t ) ⁢ ( x , x 0 ) ] / ∂ ⁡ t (in step (vi)). As is evident from (2.2), a lower bound on Ric f ⁡ ( g ) is a weaker assumption than one on Ric f m ⁡ ( g ) for m < ∞ by virtue of Ric f m ≥ k ⁢ g implying Ric f ≥ k ⁢ g but not vice versa.

Remark 3.3.

In the static case ∂ ⁡ g / ∂ ⁡ t ≡ 0 and ∂ ⁡ f / ∂ ⁡ t ≡ 0 , we set 𝗁 = 0 and then 𝖪 = 𝗄 . As such, Theorem 3.1 also gives local gradient estimates for positive solutions of (1.3) on the static metric measure space ( M , g , d ⁢ μ ) with Ric f m ⁡ ( g ) ≥ - 𝗄 m ⁢ g ; see Section 5.

If u is a positive bounded solution to (1.3) and the lower bounds Ric f m ⁡ ( g ) ≥ - 𝗄 m ⁢ g and ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ 𝗁 ⁢ g in Theorem 3.1 are global on M × [ 0 , T ] , then, by passing to R ↗ ∞ , we have the following global counterpart of (3.1).

Corollary 3.4.

Under the assumptions of Theorem 3.1 and the global bounds Ric f m ⁡ ( g ) ≥ - k m ⁢ g and ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ h ⁢ g on M × [ 0 , T ] with h , k m ≥ 0 , if u is a positive bounded solution to (1.3), there exists C > 0 such that for 0 < t ≤ T and x ∈ M ,

(3.2) | ∇ ⁡ u | u + 1 ≤ C ⁢ { ( 𝖪 + 1 t ) ⁢ ( sup M × [ 0 , T ] ⁡ u ) + sup M × [ 0 , T ] ⁡ 𝖱 F ⁢ ( u ) } .

Note that the boundedness of u does not necessarily imply that the right-hand side of (3.2) is always finite; however, this is not an issue for the estimate itself (see Section 5 for more on this). The proof of Theorem 3.1 and all the necessary tools span Section 4. In Sections 5 and 6, we give some interesting consequences of the local gradient estimate (3.1) that includes global elliptic and parabolic estimates for positive solutions, a curvature free estimate for the case when M is closed and a number of important Liouville type results on the elliptic (non-evolutionary) counterpart of (1.3).

4 Proof of the main estimate and Theorem 3.1

4.1 Parabolic lemmas

Before proceeding onto the proof of Theorem 3.1, we present a chain of lemmas that will be used in the course of the argument. First we establish some parabolic estimates under the flow on auxiliary functions involved in the proof. Note that hereafter for convenience we abbreviate the metric inner product by writing ∇ ⁡ h 1 ⁢ ∇ ⁡ h 2 = 〈 ∇ ⁡ h 1 , ∇ ⁡ h 2 〉 . We also write h t = ∂ ⁡ h / ∂ ⁡ t and as before Δ f ⁢ h = Δ ⁢ h - 〈 ∇ ⁡ f , ∇ ⁡ h 〉 .

Lemma 4.1.

Let u be a positive solution to (1.3) and set h = u . Then h is a positive solution to the equation

(4.1) h t - Δ f ⁢ h - | ∇ ⁡ h | 2 h = F ⁢ ( h 2 ) 2 ⁢ h .

Proof.

A straightforward calculation with h = u gives

2 ⁢ h t = u t u ,

2 ⁢ ∇ ⁡ h = ∇ ⁡ u u ,

2 ⁢ Δ ⁢ h = [ Δ ⁢ u - | ∇ ⁡ u | 2 2 ⁢ u ] / u .

Moreover, it is easily seen that

| ∇ ⁡ u | u = | ∇ ⁡ log ⁡ u | = 2 ⁢ | ∇ ⁡ log ⁡ u | = 2 ⁢ | ∇ ⁡ u | u .

Hence referring to (1.3) and (1.4) and putting the above pieces together, we have

2 ⁢ Δ f ⁢ h = 2 ⁢ Δ ⁢ h - 2 ⁢ ∇ ⁡ f ⁢ ∇ ⁡ h

= [ Δ ⁢ u - | ∇ ⁡ u | 2 2 ⁢ u ] / u - ∇ ⁡ f ⁢ ∇ ⁡ u u

= [ Δ f ⁢ u - | ∇ ⁡ u | 2 2 ⁢ u ] / u

= [ u t - F ⁢ ( u ) - | ∇ ⁡ u | 2 2 ⁢ u ] / u

= 2 ⁢ h ⁢ h t - F ⁢ ( h 2 ) - 2 ⁢ | ∇ ⁡ h | 2 h ,

and so the conclusion follows immediately. ∎

Lemma 4.2.

Suppose g = g ⁢ ( t ) and f = f ⁢ ( t ) are of class C 2 and let h be a positive solution to (4.1). Put W = | ∇ ⁡ h | 2 / ( 1 + h ) 2 . Then

Δ f ⁢ W - W t = g t ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 + 2 ⁢ Ric f m ⁡ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 + 2 ⁢ | ∇ 2 ⁡ h 1 + h - ∇ ⁡ h ⊗ ∇ ⁡ h ( 1 + h ) 2 | 2 + 2 ⁢ [ ∇ ⁡ f ⊗ ∇ ⁡ f ] ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( m - n ) ⁢ ( 1 + h ) 2

(4.2) - 2 ⁢ ( 2 ⁢ h + 1 ) ⁢ ∇ ⁡ h ⁢ ∇ ⁡ W ( 1 + h ) ⁢ h + 2 ⁢ ( 1 + h ) ⁢ W 2 h 2 + [ ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) - 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) ( 1 + h ) ⁢ h 2 ] ⁢ W .

Proof.

By using the description W = | ∇ ⁡ h | 2 / ( 1 + h ) 2 and the conclusion of Lemma 4.1, let us proceed by explicitly calculating the left-hand side of (4.2). Towards this end, we first note that

W t = [ | ∇ ⁡ h | 2 ( 1 + h ) 2 ] t

= ( | ∇ ⁡ h | 2 ) t ( 1 + h ) 2 - 2 ⁢ | ∇ ⁡ h | 2 ⁢ h t ( 1 + h ) 3

(4.3) = - g t ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 + 2 ⁢ ∇ ⁡ h ⁢ ∇ ⁡ h t ( 1 + h ) 2 - 2 ⁢ | ∇ ⁡ h | 2 ⁢ h t ( 1 + h ) 3 ,

where we have made use of the identity ( | ∇ ⁡ h | 2 ) t = - g t ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) + 2 ⁢ ∇ ⁡ h ⁢ ∇ ⁡ h t . Similarly, for the gradient and the Laplacian of W we have

∇ ⁡ W = ∇ | ∇ h | 2 ( 1 + h ) 2 - 2 ⁢ | ∇ ⁡ h | 2 ⁢ ∇ ⁡ h ( 1 + h ) 3 ,

Δ ⁢ W = Δ ⁢ | ∇ ⁡ h | 2 ( 1 + h ) 2 - 2 ⁢ | ∇ ⁡ h | 2 ⁢ Δ ⁢ h ( 1 + h ) 3 - 4 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 3 + 6 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 4 ,

which then gives

(4.4) Δ f ⁢ W = Δ ⁢ W - ∇ ⁡ f ⁢ ∇ ⁡ W = Δ f ⁢ | ∇ ⁡ h | 2 ( 1 + h ) 2 - 2 ⁢ | ∇ ⁡ h | 2 ⁢ Δ f ⁢ h ( 1 + h ) 3 - 4 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 3 + 6 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 4 .

Now putting together the individual descriptions of W t and Δ f ⁢ W from (4.3) and (4.4), respectively, and taking into account the relevant cancellations, we have

Δ f ⁢ W - W t = Δ f ⁢ | ∇ ⁡ h | 2 ( 1 + h ) 2 + g t ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 - 2 ⁢ ∇ ⁡ h ( 1 + h ) 2 ⁢ ∇ ⁡ ( Δ f ⁢ h + | ∇ ⁡ h | 2 h + F ⁢ ( h 2 ) h )

+ 2 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 3 ⁢ h + 2 ⁢ | ∇ ⁡ h | 2 ( 1 + h ) 3 ⁢ F ⁢ ( h 2 ) h - 4 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 3 + 6 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 4 ,

or, after doing some algebra,

Δ f ⁢ W - W t = Δ f ⁢ | ∇ ⁡ h | 2 ( 1 + h ) 2 - 2 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 2 + g t ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2

- 2 ⁢ ∇ ⁡ h ( 1 + h ) 2 ⁢ ( ∇ | ∇ h | 2 h - | ∇ ⁡ h | 2 ⁢ ∇ ⁡ h h 2 + 2 ⁢ h ⁢ F ′ ⁢ ( h 2 ) ⁢ ∇ ⁡ h 2 ⁢ h - F ⁢ ( h 2 ) ⁢ ∇ ⁡ h 2 ⁢ h 2 )

+ 2 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 3 ⁢ h + 2 ⁢ | ∇ ⁡ h | 2 ( 1 + h ) 3 ⁢ F ⁢ ( h 2 ) 2 ⁢ h - 4 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 3 + 6 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 4 .

Hence by using the weighted Bochner–Weitzenböck formula (2.3), we can write

Δ f ⁢ W - W t = 2 ⁢ Ric f m ⁡ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 + 2 ⁢ | ∇ 2 ⁡ h | 2 ( 1 + h ) 2 + g t ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 + 2 ⁢ [ ∇ ⁡ f ⊗ ∇ ⁡ f ] ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( m - n ) ⁢ ( 1 + h ) 2

- 2 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 2 ⁢ h + 2 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 2 ⁢ h 2 - 2 ⁢ | ∇ ⁡ h | 2 ⁢ F ′ ⁢ ( h 2 ) ( 1 + h ) 2 + | ∇ ⁡ h | 2 ⁢ F ⁢ ( h 2 ) ( 1 + h ) 2 ⁢ h 2

+ 2 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 3 ⁢ h + | ∇ ⁡ h | 2 ( 1 + h ) 3 ⁢ F ⁢ ( h 2 ) h - 4 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 3 + 6 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 4 ,

or, upon rearranging terms,

Δ f ⁢ W - W t = 2 ⁢ | ∇ 2 ⁡ h 1 + h - ∇ ⁡ h ⊗ ∇ ⁡ h ( 1 + h ) 2 | 2 + 2 ⁢ Ric f m ⁡ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 + g t ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( 1 + h ) 2 + 2 ⁢ [ ∇ ⁡ f ⊗ ∇ ⁡ f ] ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( m - n ) ⁢ ( 1 + h ) 2

- 2 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 2 ⁢ h - 2 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 3 + 4 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 4 + 2 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 2 ⁢ h 2 + 2 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 3 ⁢ h

+ | ∇ ⁡ h | 2 ⁢ F ⁢ ( h 2 ) ( 1 + h ) 2 ⁢ h 2 + | ∇ ⁡ h | 2 ⁢ F ⁢ ( h 2 ) ( 1 + h ) 3 ⁢ h - 2 ⁢ | ∇ ⁡ h | 2 ⁢ F ′ ⁢ ( h 2 ) ( 1 + h ) 2 .

Now by substituting and reverting to W in the terms involving | ∇ ⁡ h | 2 in the last two lines, this results in

- 2 ⁢ ∇ ⁡ h ⁢ ∇ ⁡ W h - 2 ⁢ ∇ ⁡ h ⁢ ∇ ⁡ W 1 + h + 2 ⁢ ( 1 + h ) 2 h 2 ⁢ W 2 - 2 ⁢ ( 1 + h ) h ⁢ W 2

+ W ⁢ F ⁢ ( h 2 ) h 2 + W ⁢ F ⁢ ( h 2 ) ( 1 + h ) ⁢ h - 2 ⁢ W ⁢ F ′ ⁢ ( h 2 ) ,

which is the desired conclusion. Note that here we have made use of the relation

2 ⁢ ∇ ⁡ h ⁢ ∇ ⁡ W 1 + h = 2 ∇ h ∇ | ∇ h | 2 ( 1 + h ) 3 - 4 ⁢ | ∇ ⁡ h | 4 ( 1 + h ) 4 .

This therefore completes the proof. ∎

Lemma 4.3.

Under the assumptions of Lemma 4.2, if g = g ⁢ ( t ) and f = f ⁢ ( t ) evolve under the ( k , m ) -super Perelman–Ricci flow (1.1), then

Δ f ⁢ W - W t ≥ - 2 ⁢ ( 2 ⁢ h + 1 ) ⁢ ∇ ⁡ h ⁢ ∇ ⁡ W ( 1 + h ) ⁢ h + 2 ⁢ ( 1 + h ) ⁢ W 2 h 2

+ [ ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) - 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) ( 1 + h ) ⁢ h 2 ] ⁢ W - 2 ⁢ k ⁢ W .

Proof.

This is a straightforward consequence of (4.2) by using the flow inequality (1.1) and

2 ⁢ [ ∇ ⁡ f ⊗ ∇ ⁡ f ] ⁢ ( ∇ ⁡ h , ∇ ⁡ h ) ( m - n ) ⁢ ( 1 + h ) 2 = 2 ⁢ 〈 ∇ ⁡ f , ∇ ⁡ h 〉 2 ( m - n ) ⁢ ( 1 + h ) 2 ≥ 0 ,

as desired. ∎

The proof of Theorem 3.1 uses Lemma 4.3 and a space-time localization using suitable cut-off functions whose main properties appear in Lemma 4.4 below. (For more on the method, its background and underlying ideas, see [14, 23, 2, 6, 30] and the references therein. For more discussions on the use of cut-off techniques and localization in deriving various estimates in PDEs, see also [25, 26].) Now fix R , T > 0 and then τ ∈ ( 0 , T ] . Denoting by ϱ ⁢ ( x , t ) = d g ⁢ ( t ) ⁢ ( x , x 0 ) the geodesic radial variable at time t with respect to the reference point x 0 and 0 ≤ t ≤ T , we write

(4.5) ψ ⁢ ( x , t ) = ψ ¯ ⁢ ( ϱ ⁢ ( x , t ) , t )

for a smooth cut-off function supported in the compact set ℬ R , T ⊂ M × [ 0 , T ] . The function ψ ¯ = ψ ¯ ⁢ ( ϱ , t ) appearing on the right-hand side of (4.5) is one granted by and described in the following statement (see [2, 6, 23, 31]).

Lemma 4.4.

Given τ ∈ ( 0 , T ] , there exists a smooth function ψ ¯ : [ 0 , ∞ ) × [ 0 , T ] → R such that the following properties hold:

supp ⁡ ψ ¯ ⁢ ( ϱ , t ) ⊂ [ 0 , R ] × [ 0 , T ] , and 0 ≤ ψ ¯ ⁢ ( ϱ , t ) ≤ 1 in [ 0 , R ] × [ 0 , T ] .
ψ ¯ = 1 in [ 0 , R 2 ] × [ τ , T ] and ∂ ⁡ ψ ¯ / ∂ ⁡ ϱ = 0 in [ 0 , R 2 ] × [ 0 , T ] , respectively.
| ∂ ⁡ ψ ¯ / ∂ ⁡ t | ≤ c ⁢ ψ ¯ 1 / 2 / τ on [ 0 , ∞ ) × [ 0 , T ] for some c > 0 , and ψ ¯ ⁢ ( ϱ , 0 ) = 0 for all ϱ ∈ [ 0 , ∞ ) .
- c a ⁢ ψ ¯ a / R ≤ ∂ ⁡ ψ ¯ / ∂ ⁡ ϱ ≤ 0 and | ∂ 2 ⁡ ψ ¯ / ∂ ⁡ ϱ 2 | ≤ c a ⁢ ψ ¯ a / R 2 hold on [ 0 , ∞ ) × [ 0 , T ] for every 0 < a < 1 and some c a > 0 .

4.2 Proof of Theorem 3.1

Let ψ be as described in (4.5) with ψ ¯ as in Lemma 4.4. Note that here we have fixed R ≥ 2 , T > 0 and 0 < τ ≤ T . We will show that (3.1) holds for all ( x , τ ) with d ⁢ ( x , x 0 ; τ ) ≤ R 2 , and then the arbitrariness of τ will grant the assertion for all ( x , t ) in ℬ R / 2 , T with t ≠ 0 . Now by proceeding forward, a straightforward calculation gives Δ f ⁢ ( ψ ⁢ W ) = ψ ⁢ Δ f ⁢ W + 2 ⁢ ∇ ⁡ ψ ⁢ ∇ ⁡ W + W ⁢ Δ f ⁢ ψ . So when combined with ( ψ ⁢ W ) t = ψ ⁢ W t + W ⁢ ψ t , we can write

( Δ f - ∂ ∂ ⁡ t ) ⁢ ( ψ ⁢ W ) = ψ ⁢ ( Δ f - ∂ ∂ ⁡ t ) ⁢ W + 2 ⁢ ∇ ⁡ ψ ⁢ ∇ ⁡ W + W ⁢ ( Δ f - ∂ ∂ ⁡ t ) ⁢ ψ .

At this stage we refer to the conclusion in Lemma 4.3 and, by substitution and making note of the non-negativity of ψ, we write

( Δ f - ∂ ∂ ⁡ t ) ⁢ ( ψ ⁢ W ) ≥ - 2 ⁢ ( 2 ⁢ h + 1 ) ⁢ ψ ⁢ ∇ ⁡ h ⁢ ∇ ⁡ W ( 1 + h ) ⁢ h + 2 ⁢ ( 1 + h ) ⁢ ψ ⁢ W 2 h 2 + 2 ⁢ ∇ ⁡ ψ ⁢ ∇ ⁡ W

+ [ ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) - 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) ( 1 + h ) ⁢ h 2 ] ⁢ ψ ⁢ W

(4.6) + W ⁢ ( Δ f - ∂ ∂ ⁡ t - 2 ⁢ k ) ⁢ ψ .

Next by utilizing the two basic vector identities

ψ ⁢ ∇ ⁡ h ⁢ ∇ ⁡ W = ∇ ⁡ h ⁢ ∇ ⁡ ( ψ ⁢ W ) - ( ∇ ⁡ h ⁢ ∇ ⁡ ψ ) ⁢ W and ∇ ⁡ ψ ⁢ ∇ ⁡ W = ( ∇ ⁡ ψ / ψ ) ⁢ ∇ ⁡ ( ψ ⁢ W ) - ( | ∇ ⁡ ψ | 2 ψ ) ⁢ W

and upon substituting in (4.6), we have

( Δ f - ∂ ∂ ⁡ t ) ⁢ ( ψ ⁢ W ) ≥ 2 ⁢ ( 2 ⁢ h + 1 ) ⁢ [ ( ∇ ⁡ h ⁢ ∇ ⁡ ψ ) ⁢ W - ∇ ⁡ h ⁢ ∇ ⁡ ( ψ ⁢ W ) ( 1 + h ) ⁢ h ]

+ 2 ⁢ ∇ ⁡ ψ ψ ⁢ ∇ ⁡ ( ψ ⁢ W ) - 2 ⁢ | ∇ ⁡ ψ | 2 ψ ⁢ W + 2 ⁢ ( 1 + h ) ⁢ ψ ⁢ W 2 h 2

(4.7) + [ ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) - 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) ( 1 + h ) ⁢ h 2 ] ⁢ ψ ⁢ W + W ⁢ ( Δ f - ∂ ∂ ⁡ t - 2 ⁢ k ) ⁢ ψ .

Assume now that the localized function ψ ⁢ W attains its maximum value on the compact set ℬ R , T at ( x 1 , t 1 ) . We suppose without loss of generality that x 1 is not on the cut-locus of M by Calabi’s argument [14]. We also assume that ( ψ ⁢ W ) ⁢ ( x 1 , t 1 ) > 0 , as otherwise the desired estimate becomes trivial with W ⁢ ( x , τ ) ≤ 0 whenever d ⁢ ( x , x 0 ; τ ) ≤ R 2 . Thus in particular t 1 > 0 by (iii) and at ( x 1 , t 1 ) we have Δ f ⁢ ( ψ ⁢ W ) ≤ 0 , ( ψ ⁢ W ) t ≥ 0 and ∇ ⁡ ( ψ ⁢ W ) = 0 . Therefore, (4.7) after taking into account all the necessary cancellations implies that

2 ⁢ ( 1 + h ) ⁢ ψ ⁢ W 2 h 2 ≤ - 2 ⁢ ( 2 ⁢ h + 1 ) ⁢ ( ∇ ⁡ h ⁢ ∇ ⁡ ψ ) ⁢ W ( 1 + h ) ⁢ h + 2 ⁢ | ∇ ⁡ ψ | 2 ψ ⁢ W

- [ ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) - 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) ( 1 + h ) ⁢ h 2 ] ⁢ ψ ⁢ W - W ⁢ ( Δ f - ∂ ∂ ⁡ t - 2 ⁢ k ) ⁢ ψ

at the point ( x 1 , t 1 ) . After multiplying through by h 2 / 2 ⁢ ( 1 + h ) , this can be rewritten as

ψ ⁢ W 2 ≤ - ( 2 ⁢ h + 1 ) ⁢ ( ∇ ⁡ h ⁢ ∇ ⁡ ψ ) ⁢ W ⁢ h ( 1 + h ) 2 + | ∇ ⁡ ψ | 2 ⁢ W ⁢ h 2 ψ ⁢ ( 1 + h ) + [ 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) 2 ] ⁢ ψ ⁢ W

(4.8) + k ⁢ ψ ⁢ W ⁢ h 2 1 + h - W ⁢ h 2 ⁢ Δ f ⁢ ψ 2 ⁢ ( 1 + h ) + W ⁢ h 2 ⁢ ψ t 2 ⁢ ( 1 + h ) .

The plan is now to exploit the above inequality and the maximal characterization of ( x 1 , t 1 ) to establish the required estimate at the space-time point ( x , τ ) . Towards this end, we proceed by considering two cases depending as to whether d ⁢ ( x 1 , x 0 ; t 1 ) ≥ 1 or d ⁢ ( x 1 , x 0 ; t 1 ) ≤ 1 . Let us consider the first case. Here we apply the properties (i)–(iv) of ψ as listed in Lemma 4.4, and the Cauchy–Schwarz and Young inequalities, respectively, to obtain suitable upper bounds for each of the six terms on the right-hand side of (4.8).

(i) For the first term, by recalling the definition of W, noting 0 < ( 2 ⁢ h + 1 ) / ( 1 + h ) ≤ 2 and using basic inequalities, we can write

- ( 2 ⁢ h + 1 ) ⁢ ( ∇ ⁡ h ⁢ ∇ ⁡ ψ ) ⁢ W ⁢ h ( 1 + h ) 2 ≤ ( 2 ⁢ h + 1 ) ⁢ | ∇ ⁡ h | ⁢ | ∇ ⁡ ψ | ⁢ W ⁢ h ( 1 + h ) 2

≤ ( 2 ⁢ h + 1 ) ⁢ h 1 + h ⁢ | ∇ ⁡ h | 1 + h ⁢ | ∇ ⁡ ψ | ⁢ W

≤ 2 ⁢ | ∇ ⁡ ψ | ⁢ W 3 / 2 ⁢ ( sup ℬ R , T ⁡ h )

≤ 1 8 ⁢ ψ ⁢ W 2 + C ⁢ ( | ∇ ⁡ ψ | ⁢ ( sup ℬ R , T ⁡ h ) ψ 3 / 4 ) 4

≤ 1 8 ⁢ ψ ⁢ W 2 + C R 4 ⁢ ( sup ℬ R , T ⁡ h ) 4 .

(ii) For the second term, noting 1 / ( 1 + h ) ≤ 1 and proceeding in a similar way, we have

| ∇ ⁡ ψ | 2 ψ ⁢ W ⁢ h 2 1 + h ≤ ψ ⁢ W ⁢ | ∇ ⁡ ψ | 2 ψ 3 / 2 ⁢ ( sup ℬ R , T ⁡ h ) 2

≤ 1 8 ⁢ ψ ⁢ W 2 + C ⁢ ( | ∇ ⁡ ψ | 2 ψ 3 / 2 ) 2 ⁢ ( sup ℬ R , T ⁡ h ) 4

≤ 1 8 ⁢ ψ ⁢ W 2 + C R 4 ⁢ ( sup ℬ R , T ⁡ h ) 4 .

(iii) For the third term, noting 0 ≤ ψ ≤ 1 and taking positive parts, we can write

[ 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) 2 ] ⁢ ψ ⁢ W ≤ ψ ⁢ W ⁢ [ 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) 2 ] +

≤ 1 8 ⁢ ψ ⁢ W 2 + C ⁢ sup ℬ R , T ⁡ 𝖱 F 2 ⁢ ( u ) ,

where

𝖱 F ⁢ ( u ) = { 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) 2 } + .

(iv) For the fourth term we can write

k ⁢ ψ ⁢ W ⁢ h 2 1 + h ≤ k + ⁢ ψ ⁢ W ⁢ ψ 1 + h ⁢ ( sup ℬ R , T ⁡ h ) 2 ≤ 1 8 ⁢ ψ ⁢ W 2 + C ⁢ k + 2 ⁢ ( sup ℬ R , T ⁡ h ) 4 .

(v) For the fifth term we treat the cases m = ∞ and n ≤ m < ∞ separately. Before that however, we note that in view of the relation Ric f ⁡ ( g ) = Ric f m ⁡ ( g ) + [ ∇ ⁡ f ⊗ ∇ ⁡ f ] / ( m - n ) (see (2.2) and Remark 3.2), the estimate obtained in the first case under a lower bound on Ric f ⁡ ( g ) remains true also in the second case. However, as will be seen, one can take advantage of the lower bound on Ric f m ⁡ ( g ) in the second case to obtain a slightly less involved estimate (compare (4.9) with (4.10) below). Proceeding now onto the first case, we set ζ = max ⁡ ( Z Δ f , 0 ) ≥ 0 where

Z Δ f = max ( x , t ) ⁡ { Δ f ⁢ ϱ ⁢ ( x , t ) : d ⁢ ( x , x 0 ; t ) = 1 , 0 ≤ t ≤ T } .

Using Ric f ⁡ ( g ) ≥ - ( n - 1 ) ⁢ 𝗄 ⁢ g and the weighted Laplacian comparison theorem [30, Theorem 3.1], we have Δ f ⁢ ϱ ≤ ζ + ( R - 1 ) ⁢ ( n - 1 ) ⁢ 𝗄 for ϱ ≥ 1 . From ψ being radial with ψ ¯ ϱ ≤ 0 it then follows that

Δ f ⁢ ψ = ψ ¯ ϱ ⁢ ϱ ⁢ | ∇ ⁡ ϱ | 2 + ψ ¯ ϱ ⁢ Δ f ⁢ ϱ ≥ ψ ¯ ϱ ⁢ ϱ + ψ ¯ ϱ ⁢ [ ζ + ( R - 1 ) ⁢ ( n - 1 ) ⁢ 𝗄 ] ,

and so

- Δ f ⁢ ψ ≤ | ψ ¯ ϱ ⁢ ϱ | + [ ζ + ( R - 1 ) ⁢ ( n - 1 ) ⁢ 𝗄 ] ⁢ | ψ ¯ ϱ | .

Hence, we can write

h 2 ⁢ W 1 + h ⁢ ( - Δ f ⁢ ψ ) ≤ h 2 ⁢ ψ ¯ ⁢ W 1 + h ⁢ ( | ψ ¯ ϱ ⁢ ϱ | ψ ¯ + [ ζ + ( R - 1 ) ⁢ ( n - 1 ) ⁢ 𝗄 ] ⁢ | ψ ¯ ϱ | ψ ¯ )

≤ ψ ¯ ⁢ W ⁢ ( | ψ ¯ ϱ ⁢ ϱ | ψ ¯ + [ ζ + ( R - 1 ) ⁢ ( n - 1 ) ⁢ 𝗄 ] ⁢ | ψ ¯ ϱ | ψ ¯ ) ⁢ ( sup ℬ R , T ⁡ h ) 2

≤ 1 8 ⁢ ψ ¯ ⁢ W 2 + C ⁢ ( | ψ ¯ ϱ ⁢ ϱ | 2 ψ ¯ + ζ 2 ⁢ | ψ ¯ ϱ | 2 ψ ¯ + n 2 ⁢ 𝗄 2 ⁢ R 2 ⁢ | ψ ¯ ϱ | 2 ψ ¯ ) ⁢ ( sup ℬ R , T ⁡ h ) 4

(4.9) ≤ 1 8 ⁢ ψ ⁢ W 2 + C ⁢ n 2 R 4 ⁢ ( 1 + ζ 2 ⁢ R 2 + 𝗄 2 ⁢ R 4 ) ⁢ ( sup ℬ R , T ⁡ h ) 4 .

In the second case, using Ric f m ⁡ ( g ) ≥ - ( m - 1 ) ⁢ 𝗄 ⁢ g , we have Δ f ⁢ ϱ ≤ ( m - 1 ) ⁢ 𝗄 ⁢ coth ⁡ ( 𝗄 ⁢ ϱ ) (see [30]). Therefore, again by virtue of ψ being radial and ψ ¯ ϱ ≤ 0 , it follows that

Δ f ⁢ ψ = ψ ¯ ϱ ⁢ ϱ ⁢ | ∇ ⁡ ϱ | 2 + ψ ¯ ϱ ⁢ Δ f ⁢ ϱ ≥ ψ ¯ ϱ ⁢ ϱ + ( m - 1 ) ⁢ ψ ¯ ϱ ⁢ 𝗄 ⁢ coth ⁡ ( 𝗄 ⁢ ϱ ) .

By using the bound

𝗄 ⁢ coth ⁡ ( 𝗄 ⁢ ϱ ) ≤ 𝗄 ⁢ coth ⁡ ( 𝗄 ⁢ R 2 ) ≤ 2 + 𝗄 ⁢ R R for ⁢ R 2 ≤ ϱ ≤ R

(here we are making use of v ⁢ coth ⁡ v ≤ 1 + v and the monotonicity of coth ⁡ v for v > 0 ) and noting that ψ ¯ ϱ ≡ 0 for 0 ≤ ϱ ≤ R 2 , it then follows that

- Δ f ⁢ ψ ≤ - [ ψ ¯ ϱ ⁢ ϱ + ( m - 1 ) ⁢ ψ ¯ ϱ ⁢ 𝗄 ⁢ coth ⁡ ( 𝗄 ⁢ R 2 ) ] ≤ | ψ ¯ ϱ ⁢ ϱ | + ( m - 1 ) ⁢ ( 2 R + 𝗄 ) ⁢ | ψ ¯ ϱ | .

Therefore, we can write

h 2 ⁢ W 1 + h ⁢ ( - Δ f ⁢ ψ ) ≤ ψ ¯ ⁢ W ⁢ ( | ψ ¯ ϱ ⁢ ϱ | ψ ¯ + ( m - 1 ) ⁢ ( 2 R + 𝗄 ) ⁢ | ψ ¯ ϱ | ψ ¯ ) ⁢ ( sup ℬ R , T ⁡ h ) 2

≤ 1 8 ⁢ ψ ¯ ⁢ W 2 + C ⁢ ( | ψ ¯ ϱ ⁢ ϱ | 2 ψ ¯ + m 2 ⁢ ( 1 R 2 + 𝗄 ) ⁢ | ψ ¯ ϱ | 2 ψ ¯ ) ⁢ ( sup ℬ R , T ⁡ h ) 4

(4.10) ≤ 1 8 ⁢ ψ ⁢ W 2 + C ⁢ m 2 R 4 ⁢ ( 1 + 𝗄 ⁢ R 2 ) ⁢ ( sup ℬ R , T ⁡ h ) 4 .

(vi) For the sixth term we first estimate ψ t as follows: For d ⁢ ( x 1 , x 0 ; t ) ≤ R and fixed t > 0 let

γ = γ ⁢ ( s ) : [ 0 , d ] → M

be a minimal geodesic with respect to g ⁢ ( t ) connecting x 0 = γ ⁢ ( 0 ) to x 1 = γ ⁢ ( d ) . Then a standard calculation utilizing ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ 𝗁 ⁢ g in ℬ R , T with 𝗁 ≥ 0 as in the theorem gives

∂ ∂ ⁡ t ⁢ ϱ ⁢ ( x 1 , t ) = ∂ ∂ ⁡ t ⁢ d ⁢ ( x 1 , x 0 ; t )

= ∂ ∂ ⁡ t ⁢ ∫ 0 d | γ ′ ⁢ ( s ) | g ⁢ ( t ) ⁢ 𝑑 s

= ∫ 0 d ( ∂ ⁡ g / ∂ ⁡ t ) ⁢ ( γ ′ , γ ′ ) 2 ⁢ | γ ′ | g ⁢ ( t ) ⁢ 𝑑 s

≥ ∫ 0 d - 𝗁 ⁢ | γ ′ | g ⁢ ( t ) ⁢ d ⁢ s

≥ - 𝗁 ⁢ ϱ ⁢ ( x 1 , t ) ≥ - 𝗁 ⁢ R .

Thus referring to (4.5) and applying the properties of ψ ¯ as listed in Lemma 4.4, we deduce that

ψ t = ψ ¯ t + ψ ¯ ϱ ⁢ ϱ t ≤ ψ ¯ t - ψ ¯ ϱ ⁢ 𝗁 ⁢ R ≤ | ψ ¯ t | + | ψ ¯ ϱ | ⁢ 𝗁 ⁢ R ≤ C ⁢ ( 1 τ + 𝗁 ) ⁢ ψ .

Now with the aid of this estimate of ψ t , we can proceed onto the last term and write

h 2 ⁢ W 1 + h ⁢ ψ t = ψ ⁢ h 2 ⁢ W 1 + h ⁢ ψ t ψ

≤ C ⁢ ψ ⁢ h 2 ⁢ W 1 + h ⁢ ( 1 τ + 𝗁 )

≤ C ⁢ ψ ⁢ W ⁢ ( 1 τ + 𝗁 ) ⁢ h 2

≤ 1 8 ⁢ ψ ⁢ W 2 + C ⁢ ( 1 τ 2 + 𝗁 2 ) ⁢ ( sup ℬ R , T ⁡ h ) 4 .

Completing the estimate of the individual terms on the right-hand side of (4.8), by inserting these estimates back and after basic considerations and adjusting the constants upon noting R ≥ 2 , we arrive at the following bound at ( x 1 , t 1 ) :

(4.11) ψ ⁢ W 2 ≤ 3 4 ⁢ ψ ⁢ W 2 + C ⁢ { 1 R 4 ⁢ ( 1 + ζ 2 ⁢ R 2 + R 4 τ 2 + 𝖪 2 ⁢ R 4 ) ⁢ ( sup ℬ R , T ⁡ h ) 4 + sup ℬ R , T ⁡ 𝖱 F 2 ⁢ ( u ) } ,

where 𝖪 = 𝗁 2 + 𝗄 2 . Referring to the expression on the right-hand side in (4.11), note that in the case resulting from (4.10) in case two in (v), we can remove the term ζ 2 ⁢ R 2 . Now invoking the maximal characterization of ( x 1 , t 1 ) , we have for all ( x , t ) ∈ ℬ R , T the inequalities

ψ 2 ⁢ W 2 ⁢ ( x , t ) ≤ ψ 2 ⁢ W 2 ⁢ ( x 1 , t 1 ) ≤ ψ ⁢ W 2 ⁢ ( x 1 , t 1 )

and

ψ 2 ⁢ W 2 ⁢ ( x , t ) ≤ C ⁢ { 1 R 4 ⁢ ( 1 + ζ 2 ⁢ R 2 + R 4 τ 2 + 𝖪 2 ⁢ R 4 ) ⁢ ( sup ℬ R , T ⁡ h ) 4 + sup ℬ R , T ⁡ 𝖱 F 2 ⁢ ( u ) } .

Since ψ ⁢ ( x , τ ) = 1 in d ⁢ ( x , x 0 ; τ ) ≤ R 2 and by the definition W = | ∇ ⁡ h | 2 / ( 1 + h ) 2 , upon recalling h = u and | ∇ ⁡ h | = | ∇ ⁡ u | , this gives

| ∇ ⁡ u | 1 + u ≤ C ⁢ { ( 1 R + ζ R + 1 τ + 𝖪 ) ⁢ ( sup ℬ R , T ⁡ u ) + sup ℬ R , T ⁡ 𝖱 F ⁢ ( u ) } .

We now move to the second case and consider d ⁢ ( x 1 , x 0 ; t 1 ) ≤ 1 . As by Lemma 4.4 (ii), ψ is a constant function in the space direction (when d ⁢ ( x , x 0 ; t ) ≤ R 2 , t ∈ [ 0 , T ] , with R ≥ 2 ). By referring to (4.8), all the terms involving spatial derivatives of ψ vanish or simplify (including ψ t = ψ ¯ t ), and so we have at the point ( x 1 , t 1 ) ,

W ≤ { 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) ⁢ h 2 + ( k + 1 2 ⁢ ψ t ψ ) } ⁢ h 2 1 + h

≤ { [ 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) ⁢ h 2 ] + + ( k + + 1 2 ⁢ | ψ ¯ t | ψ ¯ ) } ⁢ h 2 1 + h

≤ C ⁢ { sup ℬ R , T ⁡ 𝖱 F ⁢ ( u ) + k + + 1 / τ h 2 } .

By noting W ⁢ ( x , τ ) = ψ ⁢ W ⁢ ( x , τ ) ≤ ψ ⁢ W ⁢ ( x 1 , t 1 ) ≤ W ⁢ ( x 1 , t 1 ) due to ψ ⁢ ( x 1 , t 1 ) ≤ 1 , ψ ⁢ ( x , τ ) = 1 when d ⁢ ( x , x 0 ; τ ) ≤ R 2 , the above and the arbitrariness of τ > 0 easily lead to a special case of (3.1). We have now established the estimate in both cases, and so the proof is complete.

5 Applications to special nonlinearities F and Liouville theorems

In this section, we present some corollaries of Theorem 3.1 to the static case (see Remark 3.3). Here ( M , g , e - f ⁢ d ⁢ v ) is a smooth metric measure space where the metric g and the potential f are time independent whilst Ric f m ⁡ ( g ) ≥ - 𝗄 m ⁢ g in ℬ R with 𝗄 m ≥ 0 and R ≥ 2 . For clarification, ℬ R ⊂ M is the geodesic ball with center x 0 and radius R > 0 . We consider positive bounded solutions to the elliptic equation Δ f ⁢ u + F ⁢ ( u ) = 0 . Since u is a time independent solution of (1.3), setting t ↗ ∞ in (3.1) and noting that the constants do not depend on t gives the corresponding estimate for x ∈ ℬ R / 2 :

(5.1) | ∇ ⁡ u | 1 + u = | ∇ ⁡ u | / 2 u + u ≤ C ⁢ { ( 1 R + ζ R + 𝗄 ) ⁢ ( sup ℬ R ⁡ u ) + sup ℬ R ⁡ 𝖱 F ⁢ ( u ) } .

As g t ≡ 0 , we have set 𝗁 = 0 and so 𝖪 = 𝗄 whilst

ζ = [ Z Δ f ] + = [ max { x : d ⁢ ( x , x 0 ) = 1 } ⁡ Δ f ⁢ r ] +

and

𝖱 F ⁢ ( u ) = { 2 ⁢ u ⁢ ( 1 + u ) ⁢ F ′ ⁢ ( u ) - ( 1 + 2 ⁢ u ) ⁢ F ⁢ ( u ) 2 ⁢ ( 1 + u ) 2 } + .

Now subject to the global bound Ric f m ⁡ ( g ) ≥ - 𝗄 m ⁢ g on M, setting R ↗ ∞ in (5.1) gives the global gradient estimate for positive bounded solutions u to the equation Δ f ⁢ u + F ⁢ ( u ) = 0 , namely

(5.2) | ∇ ⁡ u | 1 + u = | ∇ ⁡ u | / 2 u + u ≤ C ⁢ { 𝗄 ⁢ ( sup M ⁡ u ) + sup M ⁡ 𝖱 F ⁢ ( u ) } .

Of particular interest here are Liouville type theorems; the underlying idea upon referring to (5.2) being that if Ric f m ⁡ ( g ) ≥ 0 (i.e., 𝗄 = 0 ) and 𝖱 F ⁢ ( u ) ≡ 0 , then combined with the above estimate these together imply that ∇ ⁡ u = 0 and so u is a constant. Below we examine this more closely in the context of certain power-like and logarithmic type nonlinearities F that arise frequently in the literature and include the following:

F ⁢ ( u ) = A ⁢ u ⁢ | log ⁡ u | α (with real α > 1 ).
F ⁢ ( u ) = A ⁢ u ⁢ ( log ⁡ u ) d (with integer d ≥ 1 ).
F ⁢ ( u ) = A ⁢ u p + B ⁢ u q (with arbitrary real exponents p , q ).
F ⁢ ( u ) = A ⁢ u p ⁢ log ⁡ u + B ⁢ u q (with arbitrary real exponents p , q ).

5.1 The case F ⁢ ( u ) = A ⁢ u ⁢ | log ⁡ u | α with α > 1

The function F here is continuously differentiable, and so, by referring to the description of 𝖱 F ⁢ ( u ) (see (1.5)), a basic calculation gives

2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) 2 = A ⁢ u ⁢ | log ⁡ u | α 2 ⁢ ( 1 + h ) 2 + A ⁢ α ⁢ u ⁢ | log ⁡ u | α - 2 ⁢ log ⁡ u 1 + h .

Therefore,

𝖱 F ⁢ ( u ) = [ 2 - 1 ⁢ A ⁢ u ⁢ | log ⁡ u | α ( 1 + h ) 2 + A ⁢ α ⁢ u ⁢ | log ⁡ u | α - 2 ⁢ log ⁡ u 1 + h ] + ,

and so as readily seen

𝖱 F ⁢ ( u ) ≤ 2 - 1 ⁢ A + ⁢ u ⁢ | log ⁡ u | α + α ⁢ u ⁢ [ A ⁢ | log ⁡ u | α - 2 ⁢ log ⁡ u ] + .

Theorem 5.1.

Let ( M , g , e - f ⁢ d ⁢ v ) be a smooth metric measure space with Ric f ⁡ ( g ) ≥ 0 and consider the equation Δ f ⁢ u + A ⁢ u ⁢ | log ⁡ u | α = 0 (with α > 1 ) on M. If A ≤ 0 , then any solution with 1 ≤ u ≤ D is a constant. Additionally if A < 0 , then necessarily u ≡ 1 .

Proof.

We have A + = 0 and [ A ⁢ | log ⁡ u | α - 2 ⁢ log ⁡ u ] + = 0 by referring to the above calculation 𝖱 F ⁢ ( u ) = 0 . Since by assumption 𝗄 = 0 , it then follows from (5.2) that ∇ ⁡ u ≡ 0 , and this leads immediately to the desired conclusion. ∎

5.2 The case F ⁢ ( u ) = A ⁢ u ⁢ ( log ⁡ u ) d with integer d ≥ 1

By comparing with Section 5.1 above, the modulus sign in F has been removed from the term log ⁡ u ; however, it is evident that due to the sign changing nature of the logarithm some restriction has to be imposed on the exponent (here an integer d ≥ 1 ) as speaking of arbitrary powers of log ⁡ u for u > 0 is in general meaningless. Now a short calculation gives

2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) = A ⁢ u ⁢ ( log ⁡ u ) d 2 ⁢ ( 1 + h ) 2 + A ⁢ d ⁢ u ⁢ ( log ⁡ u ) d - 1 1 + h ,

and so referring to (1.5), we can write

𝖱 F ⁢ ( u ) = [ A ⁢ u ⁢ ( log ⁡ u ) d 2 ⁢ ( 1 + h ) 2 + A ⁢ d ⁢ u ⁢ ( log ⁡ u ) d - 1 1 + h ] +

≤ 2 - 1 ⁢ u ⁢ [ A ⁢ ( log ⁡ u ) d ] + + d ⁢ u ⁢ [ A ⁢ ( log ⁡ u ) d - 1 ] + .

Theorem 5.2.

Let ( M , g , e - f ⁢ d ⁢ v ) be a smooth metric measure space with Ric f ⁡ ( g ) ≥ 0 and consider the equation Δ f ⁢ u + A ⁢ u ⁢ ( log ⁡ u ) d = 0 (with an integer d ≥ 1 ) on M. If A ≤ 0 , then any solution with 1 ≤ u ≤ D is a constant. Additionally if A < 0 , then necessarily u ≡ 1 .

Proof.

For A ≤ 0 and d ≥ 2 even we have [ A ⁢ ( log ⁡ u ) d ] + = 0 with [ A ⁢ ( log ⁡ u ) d - 1 ] + = 0 only when u ≥ 1 . For A ≤ 0 and d ≥ 1 odd we have [ A ⁢ ( log ⁡ u ) d - 1 ] + = 0 with [ A ⁢ ( log ⁡ u ) d ] + = 0 only when u ≥ 1 . Thus A ≤ 0 and u ≥ 1 give 𝖱 F ⁢ ( u ) = 0 . The rest is as in the proof of Theorem 5.1. ∎

5.3 The case F ⁢ ( u ) = A ⁢ u p + B ⁢ u q with real exponents p , q

A direct calculation in the spirit of those in the previous sections here gives

2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) 2 = A ⁢ u p 1 + h ⁢ [ p - 1 + 2 ⁢ h 2 ⁢ ( 1 + h ) ] + B ⁢ u q 1 + h ⁢ [ q - 1 + 2 ⁢ h 2 ⁢ ( 1 + h ) ] ,

and so from (1.5) and by considering positive parts, we have

𝖱 F ⁢ ( u ) ≤ [ A ⁢ ( 2 ⁢ p - 1 ) + 2 ⁢ h ⁢ ( p - 1 ) 2 ⁢ ( 1 + h ) 2 ] + ⁢ u p + [ B ⁢ ( 2 ⁢ q - 1 ) + 2 ⁢ h ⁢ ( q - 1 ) 2 ⁢ ( 1 + h ) 2 ] + ⁢ u q .

Now as 0 < h = u ≤ D , we have

( 2 ⁢ p - 1 ) + 2 ⁢ h ⁢ ( p - 1 ) ≥ 0 if ⁢ p ≥ 1 + 2 ⁢ D 2 + 2 ⁢ D ,

and

( 2 ⁢ q - 1 ) + 2 ⁢ h ⁢ ( q - 1 ) ≤ 0 if ⁢ q ≤ 1 2 .

Thus in particular, if A ≤ 0 , B ≥ 0 , p ≥ [ 1 + 2 ⁢ D ] / [ 2 + 2 ⁢ D ] and q ≤ 1 2 , it follows that 𝖱 F ⁢ ( u ) ≡ 0 .

Theorem 5.3.

Let ( M , g , e - f ⁢ d ⁢ v ) be a smooth metric measure space with Ric f ⁡ ( g ) ≥ 0 . Assume 0 < u ≤ D is a solution to the equation Δ f ⁢ u + A ⁢ u p + B ⁢ u q = 0 on M. If A ≤ 0 , B ≥ 0 , p ≥ [ 1 + 2 ⁢ D ] / [ 2 + 2 ⁢ D ] and q ≤ 1 2 , then u is a constant.

Note that regardless of the upper bound on u the condition p ≥ [ 1 + 2 ⁢ D ] / [ 2 + 2 ⁢ D ] is always implied by p ≥ 1 . Likewise, q ≤ ( 1 + 2 ⁢ h ) / ( 2 + 2 ⁢ h ) is implied by q ≤ 1 2 .

Proof.

From the discussion preceding the theorem, the assumptions on the exponents p , q the solution u and the coefficients A , B give 𝖱 F ≡ 0 . Hence from the global estimate (5.2) with 𝗄 = 0 , it follows that ∇ ⁡ u ≡ 0 . This immediately gives the conclusion. ∎

5.4 The case F ⁢ ( u ) = A ⁢ u p ⁢ log ⁡ u + B ⁢ u q

Here again by a direct differentiation, we have

F ′ ⁢ ( u ) = A ⁢ ( p ⁢ u p - 1 ⁢ log ⁡ u + u p - 1 ) + B ⁢ q ⁢ u q - 1 .

So referring to (1.5), we can write

𝖱 F ⁢ ( u ) = [ 2 ⁢ h 2 ⁢ ( 1 + h ) ⁢ F ′ ⁢ ( h 2 ) - ( 1 + 2 ⁢ h ) ⁢ F ⁢ ( h 2 ) 2 ⁢ ( 1 + h ) 2 ] +

= [ A ⁢ u p 1 + h ⁢ { [ ( 2 ⁢ p - 1 ) + 2 ⁢ h ⁢ ( p - 1 ) ] ⁢ log ⁡ u 2 ⁢ ( 1 + h ) + 1 } + B ⁢ [ ( 2 ⁢ q - 1 ) + 2 ⁢ h ⁢ ( q - 1 ) ] ⁢ u q 2 ⁢ ( 1 + h ) 2 ] +

≤ [ A ⁢ { [ ( 2 ⁢ p - 1 ) + 2 ⁢ h ⁢ ( p - 1 ) ] ⁢ log ⁡ u 2 ⁢ ( 1 + h ) + 1 } ] + ⁢ u p + [ B ⁢ [ ( 2 ⁢ q - 1 ) + 2 ⁢ h ⁢ ( q - 1 ) ] 2 ⁢ ( 1 + h ) ] + ⁢ u q .

Let us now set s ⁢ ( p ) = [ ( 2 ⁢ p - 1 ) + 2 ⁢ h ⁢ ( p - 1 ) ] / [ 2 ⁢ ( 1 + h ) ] . Then from the above we have

𝖱 F ≤ [ A ⁢ ( s ⁢ ( p ) ⁢ log ⁡ u + 1 ) ] + + [ B ⁢ s ⁢ ( q ) ] + .

In what follows, we look more closely at the sign of the term log ⁡ ( e ⁢ u s ⁢ ( p ) ) , or equivalently that of s ⁢ ( p ) ⁢ log ⁡ u + 1 . Towards this end, consider the function

ζ p ⁢ ( h ) = - 1 s = 2 ⁢ ( 1 + h ) ( 1 - 2 ⁢ p ) + 2 ⁢ h ⁢ ( 1 - p ) , h ≥ 0 .

As for the range 1 2 < p < 1 , the function ζ p has a singularity at h = ( 2 ⁢ p - 1 ) / [ 2 ⁢ ( 1 - p ) ] on the half-axis h > 0 and is in particular unbounded. Hereafter, we restrict attention to p outside this interval, i.e., p ≤ 1 2 or p ≥ 1 . Now it is easily seen that ζ p ′ < 0 , and so ζ p is monotonically decreasing on h ≥ 0 . Moreover, by direct evaluation, ζ p ⁢ ( 0 ) = 2 / ( 1 - 2 ⁢ p ) and lim ⁡ ζ p ⁢ ( h ) = 1 / ( 1 - p ) as h ↗ ∞ . Now when p ≤ 1 2 we have 0 < 1 / ( 1 - p ) ≤ ζ p ⁢ ( h ) ≤ 2 / ( 1 - 2 ⁢ p ) , and when p ≥ 1 we have 1 / ( 1 - p ) ≤ ζ p ⁢ ( h ) ≤ 2 / ( 1 - 2 ⁢ p ) < 0 . Thus depending on the signs of A and s ⁢ log ⁡ u + 1 , we have the following assertions:

If A ≤ 0 and s ⁢ log ⁡ u ≥ - 1 , then [ A ⁢ ( s ⁢ log ⁡ u + 1 ) ] + = 0 . Thus, in terms of p:
1. If p ≤ 1 2 , then s < 0 , and so log ⁡ u ≤ 1 / ( 1 - p ) gives log ⁡ u ≤ ζ p = - 1 s .
2. If p ≥ 1 , then s > 0 , and so log ⁡ u ≥ 2 / ( 1 - 2 ⁢ p ) gives log ⁡ u ≥ ζ p = - 1 s .
If A ≥ 0 and s ⁢ log ⁡ u ≤ - 1 , then [ A ⁢ ( s ⁢ log ⁡ u + 1 ) ] + = 0 . Thus, in terms of p:
1. If p ≤ 1 2 , then s < 0 , and so log ⁡ u ≥ 2 / ( 1 - 2 ⁢ p ) gives log ⁡ u ≥ ζ p = - 1 s .
2. If p ≥ 1 , then s > 0 , and so log ⁡ u ≤ 1 / ( 1 - p ) gives log ⁡ u ≤ ζ p = - 1 s .

As regarding the term B, we can argue exactly as in Section 5.3. By putting the above together, we have proved the following statement.

Theorem 5.4.

Let ( M , g , e - f ⁢ d ⁢ v ) be a smooth metric measure space with Ric f ⁡ ( g ) ≥ 0 . Assume u is a positive bounded solution to the equation Δ f ⁢ u + A ⁢ u p ⁢ log ⁡ u + B ⁢ u q = 0 on M. Consider the set of assumptions on A , B , u and the exponents p , q described below:

{ A ≥ 0 ⁢ and either p ≥ 1 , 0 < u ≤ exp ⁡ ( 1 1 - p ) , or p ≤ 1 2 , u ≥ exp ⁡ ( 2 1 - 2 ⁢ p ) ,

{ A ≤ 0 ⁢ and either p ≥ 1 , u ≥ exp ⁡ [ 2 1 - 2 ⁢ p ] , or p ≤ 1 2 , 0 < u ≤ exp ⁡ [ 1 1 - p ] ,

together with B ≥ 0 and q ≤ 1 2 , or B ≤ 0 and q ≥ 1 . If these conditions hold, then u must be a constant.

6 A global gradient estimate for solutions to (1.3)

We have already seen some implications of the local and global estimates to Liouville type results on the elliptic counterpart of (1.3) in the previous sections. In this section, we continue by establishing estimates on the full parabolic equation that are of a global nature whereas before the metric and potential evolved under a ( k , m ) -super Perelman–Ricci flow.

Theorem 6.1.

Assume the pair ( g ⁢ ( t ) , f ⁢ ( t ) ) t ∈ [ 0 , T ] evolves under a ( k , m ) -super Perelman–Ricci flow and let u be a positive solution to (1.3). Assume u ⁢ F ⁢ ( u ) ≤ 0 and F ′ ⁢ ( u ) ≤ 0 along u. If M is closed and k ≥ 0 , then we have the global estimate

(6.1) | ∇ u ( x , t ) | 2 ≤ 1 + 2 ⁢ k ⁢ t 2 ⁢ t [ ( sup M u | t = 0 ) 2 - u 2 ]

for all x ∈ M and 0 < t ≤ T .

Proof.

The idea is to get the estimate (6.1) out of the evolution of a suitably chosen quantity depending on the solution u. Towards this end, recalling ∂ ⁡ u / ∂ ⁡ t = Δ f ⁢ u + F ⁢ ( u ) and the flow inequality ∂ ⁡ g / ∂ ⁡ t ≥ - 2 ⁢ ( Ric f m ⁡ ( g ) + k ⁢ g ) , we can write

∂ | ∇ u | 2 ∂ ⁡ t = - g t ⁢ ( ∇ ⁡ u , ∇ ⁡ u ) + 2 ⁢ ∇ ⁡ u ⁢ ∇ ⁡ u t

≤ 2 ⁢ Ric f m ⁡ ( ∇ ⁡ u , ∇ ⁡ u ) + 2 ⁢ k ⁢ | ∇ ⁡ u | 2 + 2 ⁢ ∇ ⁡ u ⁢ ∇ ⁡ [ Δ f ⁢ u + F ⁢ ( u ) ]

(6.2) ≤ Δ f ⁢ | ∇ ⁡ u | 2 - 2 ⁢ | ∇ 2 ⁡ u | 2 + 2 ⁢ k ⁢ | ∇ ⁡ u | 2 + 2 ⁢ ∇ ⁡ u ⁢ ∇ ⁡ F ⁢ ( u ) ,

where in writing the second inequality we have made use of (2.2), the weighted Bochner–Weitzenböck formula (2.3) and [ ∇ f ⊗ ∇ f ] ( ∇ u , ∇ u ) ] / ( m - n ) = 〈 ∇ f , ∇ u 〉 2 / ( m - n ) ≥ 0 for when m < ∞ . Likewise,

(6.3) ∂ ⁡ u 2 ∂ ⁡ t = 2 ⁢ u ⁢ [ Δ f ⁢ u + F ⁢ ( u ) ] = Δ f ⁢ u 2 - 2 ⁢ | ∇ ⁡ u | 2 + 2 ⁢ u ⁢ F ⁢ ( u ) .

Now for t ≥ 0 and Z to be specified below put 𝒫 γ ⁢ [ u ] = γ ⁢ ( t ) ⁢ | ∇ ⁡ u | 2 + Z ⁢ u 2 . Here γ is a non-negative, smooth but otherwise arbitrary function. Then differentiating and making use of (6.2) and (6.3) gives

∂ ⁡ 𝒫 γ ⁢ [ u ] ∂ ⁡ t = γ ′ ⁢ | ∇ ⁡ u | 2 + γ ∂ | ∇ u | 2 ∂ ⁡ t + Z ⁢ ∂ ⁡ u 2 ∂ ⁡ t

≤ γ ⁢ [ Δ f ⁢ | ∇ ⁡ u | 2 - 2 ⁢ | ∇ 2 ⁡ u | 2 + 2 ⁢ k ⁢ | ∇ ⁡ u | 2 + 2 ⁢ ∇ ⁡ u ⁢ ∇ ⁡ F ⁢ ( u ) ] + γ ′ ⁢ | ∇ ⁡ u | 2 + Z ⁢ [ Δ f ⁢ u 2 - 2 ⁢ | ∇ ⁡ u | 2 + 2 ⁢ u ⁢ F ⁢ ( u ) ] ,

or upon rearranging terms

∂ ⁡ 𝒫 γ ⁢ [ u ] ∂ ⁡ t ≤ Δ f ⁢ [ γ ⁢ | ∇ ⁡ u | 2 + Z ⁢ u 2 ] + ( γ ′ + 2 ⁢ k ⁢ γ - 2 ⁢ Z ) ⁢ | ∇ ⁡ u | 2 - 2 ⁢ γ ⁢ | ∇ 2 ⁡ u | 2 + 2 ⁢ Z ⁢ u ⁢ F ⁢ ( u ) + 2 ⁢ γ ⁢ ∇ ⁡ u ⁢ ∇ ⁡ F ⁢ ( u ) .

Therefore, substituting for 𝒫 γ ⁢ [ u ] on the right-hand side and other considerations lead to

∂ ⁡ 𝒫 γ ⁢ [ u ] ∂ ⁡ t ≤ Δ f ⁢ 𝒫 γ ⁢ [ u ] + ( γ ′ + 2 ⁢ k ⁢ γ - 2 ⁢ Z ) ⁢ | ∇ ⁡ u | 2 + 2 ⁢ Z ⁢ u ⁢ F ⁢ ( u ) + 2 ⁢ γ ⁢ ∇ ⁡ u ⁢ ∇ ⁡ F ⁢ ( u ) .

Thus in particular upon setting Z = 1 2 and rearranging terms, we have

(6.4) ∂ ⁡ 𝒫 γ ⁢ [ u ] ∂ ⁡ t - Δ f ⁢ 𝒫 γ ⁢ [ u ] ≤ ( γ ′ + 2 ⁢ k ⁢ γ - 1 ) ⁢ | ∇ ⁡ u | 2 + u ⁢ F ⁢ ( u ) + 2 ⁢ γ ⁢ | ∇ ⁡ u | 2 ⁢ F ′ ⁢ ( u ) .

Now the function γ ⁢ ( t ) = t / ( 2 ⁢ k ⁢ t + 1 ) with k , t ≥ 0 is seen to be non-negative, smooth and to satisfy γ ⁢ ( 0 ) = 0 and γ ′ + 2 ⁢ k ⁢ γ ≤ 1 . The conclusion thus follows by an application of the weak maximum principle by virtue of M being compact. ∎

Remark 6.2.

Note that in this argument we did not use the fact that u is positive, but only u ⁢ F ⁢ ( u ) ≤ 0 to hold along u.

Specializing to the nonlinearities F introduced and studied in Section 5, we have the following corollaries of Theorem 6.1 for the positive solutions u to equation (1.3):

For the equation ∂ ⁡ u / ∂ ⁡ t = Δ f ⁢ u + A ⁢ u ⁢ | log ⁡ u | α with α > 1 we have from (6.4) that

𝒫 t ≤ Δ f ⁢ 𝒫 + A ⁢ u 2 ⁢ | log ⁡ u | α + 2 ⁢ t ⁢ A ⁢ | ∇ ⁡ u | 2 ⁢ [ | log ⁡ u | α + α ⁢ | log ⁡ u | α - 2 ⁢ log ⁡ u ] .

Therefore, if A ≤ 0 and u ≥ 1 , then 𝒫 t ≤ Δ f ⁢ 𝒫 , and so the global estimate (6.1) holds.
For the equation ∂ ⁡ u / ∂ ⁡ t = Δ f ⁢ u + A ⁢ u ⁢ ( log ⁡ u ) d with d ≥ 1 integer we have that

𝒫 t ≤ Δ f ⁢ 𝒫 + A ⁢ u 2 ⁢ ( log ⁡ u ) d + 2 ⁢ t ⁢ A ⁢ | ∇ ⁡ u | 2 ⁢ [ ( log ⁡ u ) d + d ⁢ ( log ⁡ u ) d - 1 ] .

Hence if d ≥ 2 is even, then it is exactly as in (i). If d ≥ 1 is odd, and if A ≥ 0 and 0 < u ≤ e - d , or if A ≤ 0 and u ≥ 1 , then in either case the global estimate (6.1) holds.
For the equation ∂ ⁡ u / ∂ ⁡ t = Δ f ⁢ u + A ⁢ u p + B ⁢ u q we have again from (6.4) that

𝒫 t ≤ Δ f ⁢ 𝒫 + A ⁢ u p + 1 + B ⁢ u q + 1 + 2 ⁢ t ⁢ | ∇ ⁡ u | 2 ⁢ ( A ⁢ p ⁢ u p - 1 + B ⁢ q ⁢ u q - 1 ) .

As a result if A , B ≤ 0 and p , q ≥ 0 , or if A ≤ 0 , B ≥ 0 and p ≥ 0 , q ≤ 0 , with u ≥ ( - B / A ) 1 / ( p - q ) , then 𝒫 t ≤ Δ f ⁢ 𝒫 , and so the global estimate (6.1) holds.
For the equation ∂ ⁡ u / ∂ ⁡ t = Δ f ⁢ u + A ⁢ u p ⁢ log ⁡ u + B ⁢ u q , noting (6.4) in Theorem (6.1) gives

𝒫 t ≤ Δ f ⁢ 𝒫 + A ⁢ u p + 1 ⁢ log ⁡ u + B ⁢ u q + 1 + 2 ⁢ t ⁢ | ∇ ⁡ u | 2 ⁢ [ A ⁢ u p - 1 ⁢ ( p ⁢ log ⁡ u + 1 ) + B ⁢ q ⁢ u q - 1 ] .

Thus if B ≤ 0 , q ≥ 0 and either A ≥ 0 , p ≥ 0 and 0 < u ≤ e - 1 / p , or A ≤ 0 , p ≥ 0 and u ≥ 1 , then 𝒫 t ≤ Δ f ⁢ 𝒫 , and so the global estimate (6.1) holds.

Communicated by Jan Kristensen

References

[1] A. Abolarinwa and A. Taheri, Elliptic gradient estimates for a nonlinear f-heat equation on weighted manifolds with evolving metrics and potentials, Chaos Solitons Fractals 142 (2021), Paper No. 110329. 10.1016/j.chaos.2020.110329Search in Google Scholar

[2] M. Bailesteanu, X. Cao and A. Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal. 258 (2010), no. 10, 3517–3542. 10.1016/j.jfa.2009.12.003Search in Google Scholar

[3] D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités XIX 1983/84, Lecture Notes in Math. 1123, Springer, Berlin (1985), 177–206. 10.1007/BFb0075847Search in Google Scholar

[4] D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren Math. Wiss. 348, Springer, Cham, 2012. Search in Google Scholar

[5] M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489–539. 10.1007/BF01243922Search in Google Scholar

[6] K. Brighton, A Liouville-type theorem for smooth metric measure spaces, J. Geom. Anal. 23 (2013), no. 2, 562–570. 10.1007/s12220-011-9253-5Search in Google Scholar

[7] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford Math. Monogr., Oxford University, Oxford, 2009. 10.1093/acprof:oso/9780199230723.001.0001Search in Google Scholar

[8] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci Fflow, Grad. Stud. in Math. 77, American Mathematical Society, Providence, 2006. Search in Google Scholar

[9] N. T. Dung, N. N. Khanh and Q. A. Ngô, Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces, Manuscripta Math. 155 (2018), no. 3–4, 471–501. 10.1007/s00229-017-0946-3Search in Google Scholar

[10] A. Grigor’yan, Heat Kernel and Analysis on Manifolds, AMS/IP Stud. Adv. Math. 47, American Mathematical Society, Providence, 2009. Search in Google Scholar

[11] R. S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113–126. 10.4310/CAG.1993.v1.n1.a6Search in Google Scholar

[12] B. Kleiner and J. Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. 10.2140/gt.2008.12.2587Search in Google Scholar

[13] P. Li, Geometric Analysis, Cambridge Stud. Adv. Math. 134, Cambridge University, Cambridge, 2012. 10.7202/1018492arSearch in Google Scholar

[14] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3–4, 153–201. 10.1007/BF02399203Search in Google Scholar

[15] S. Li and X.-D. Li, Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds, J. Funct. Anal. 274 (2018), no. 11, 3263–3290. 10.1016/j.jfa.2017.09.017Search in Google Scholar

[16] S. Li and X.-D. Li, On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows, Asian J. Math. 22 (2018), no. 3, 577–597. 10.4310/AJM.2018.v22.n3.a10Search in Google Scholar

[17] S. Li and X.-D. Li, W-entropy, super Perelman Ricci flows, and ( K , m ) -Ricci solitons, J. Geom. Anal. 30 (2020), no. 3, 3149–3180. 10.1007/s12220-019-00193-4Search in Google Scholar

[18] X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. (9) 84 (2005), no. 10, 1295–1361. 10.1016/j.matpur.2005.04.002Search in Google Scholar

[19] J. Lott, Some geometric properties of the Bakry–Émery–Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865–883. 10.1007/s00014-003-0775-8Search in Google Scholar

[20] F. Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), no. 8, 853–858. Search in Google Scholar

[21] G. Morrison and A. Taheri, An infinite scale of incompressible twisting solutions to the nonlinear elliptic system ℒ ⁢ [ u ; 𝖠 , 𝖡 ] = ∇ ⁡ 𝒫 and the discriminant Δ ⁢ ( h , g ) , Nonlinear Anal. 173 (2018), 209–219. 10.1016/j.na.2018.04.002Search in Google Scholar

[22] G. Perelman, The entropy formula for the Ricci Flow and its geometric application, preprint 2012, https://arxiv.org/abs/math/0211159. Search in Google Scholar

[23] P. Souplet and Q. S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. Lond. Math. Soc. 38 (2006), no. 6, 1045–1053. 10.1112/S0024609306018947Search in Google Scholar

[24] K.-T. Sturm, Super-Ricci flows for metric measure spaces, J. Funct. Anal. 275 (2018), no. 12, 3504–3569. 10.1016/j.jfa.2018.07.014Search in Google Scholar

[25] A. Taheri, Function Spaces and Partial Differential Equations. Vol. 1, Oxford Lecture Ser. Math. Appl. 40, Oxford University, Oxford, 2015. 10.1093/acprof:oso/9780198733133.001.0001Search in Google Scholar

[26] A. Taheri, Function Spaces and Partial Differential Equations. Vol. 2, Oxford Lecture Ser. Math. Appl. 41, Oxford University, Oxford, 2015. 10.1093/acprof:oso/9780198733157.001.0001Search in Google Scholar

[27] A. Taheri, Gradient estimates for a weighted Γ-nonlinear parabolic equation coupled with a super Perelman–Ricci flow and implications, submitted 2021. 10.1007/s11118-021-09969-2Search in Google Scholar

[28] C. Villani, Optimal Transport: Old and New. A Series of Comprehensive Studies in Mathematics, Grundlehren Math. Wiss, 338, Springer, Berlin, 2009. 10.1007/978-3-540-71050-9Search in Google Scholar

[29] W. Wang, Harnack inequality, heat kernel bounds and eigenvalue estimates under integral Ricci curvature bounds, J. Differential Equations 269 (2020), no. 2, 1243–1277. 10.1016/j.jde.2020.01.003Search in Google Scholar

[30] G. Wei and W. Wylie, Comparison geometry for the Bakry–Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. 10.4310/jdg/1261495336Search in Google Scholar

[31] J.-Y. Wu, Elliptic gradient estimates for a weighted heat equation and applications, Math. Z. 280 (2015), no. 1–2, 451–468. 10.1007/s00209-015-1432-9Search in Google Scholar

[32] J.-Y. Wu, Gradient estimates for a nonlinear parabolic equation and Liouville theorems, Manuscripta Math. 159 (2019), no. 3–4, 511–547. 10.1007/s00229-018-1073-5Search in Google Scholar

[33] J.-Y. Wu and P. Wu, Heat kernel on smooth metric measure spaces with nonnegative curvature, Math. Ann. 362 (2015), no. 3–4, 717–742. 10.1007/s00208-014-1146-zSearch in Google Scholar

[34] Q. S. Zhang, Sobolev Inequalities, Heat Kernels Under Ricci flow, and the Poincaré Conjecture, CRC Press, Boca Raton, 2011. 10.1201/EBK1439834596Search in Google Scholar

Received: 2020-10-14

Revised: 2021-06-02

Accepted: 2021-07-02

Published Online: 2021-09-03

Published in Print: 2023-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/acv-2020-0099

Keywords for this article

Smooth metric measure spaces; super Ricci flow; Witten Laplacian; gradient estimates; Liouville type results

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