A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.


Introduction
This paper is devoted to analyze strong maximum and comparison principles for viscosity solutions to fully nonlinear second order equations on (finite dimensional) Riemannian manifolds (M, g) of the general form The first aim of this note is to complete and extend well-known results valid in the classical Euclidean case to the more general realm of Riemannian manifolds. Our second motivation is then to lay the groundwork for investigating (one-side) Liouville-type results for fully nonlinear elliptic problems as (1.1) on general Riemannian structures. This would be mainly inspired by the recent nonlinear studies in [6], which make use of (SMP) and (SmP), and the linear Liouville properties given in [32], which are intimately connected with the stochastic completeness of the manifold (cf [32,Sect. 13.2]), see also [44,54]. These latter properties will be matter of a future research by the authors and we believe this note would be a starting point to address these issues. Before stating our main results, we begin with a glimpse on the literature on maximum principles, starting with (SMP)-(SmP) and concluding with (SCP).
The (SMP) and (SmP) for linear equations with A uniformly elliptic, b bounded and continuous and c nonnegative and bounded, date back to Hopf and it is a consequence of his Boundary Point Lemma (see e.g. [31,50,53]). A refinement of this procedure was implemented by E. Calabi for semicontinuous viscosity solutions to linear equations [23]. Instead, the literature on quasi-linear problems in R n , mostly modeled on p-Laplacian operators, is huge and we refer the interested reader to [53] for a comprehensive exposition and [52,Thm 8.1] for maximum principles under mild assumptions on the quasi-linear operator.
Let us now mention related results for fully nonlinear problems close to our model PDEs. The (SMP) and (SmP) were found by Caffarelli and Cabré [21,Prop 4.9] as a consequence of the weak Harnack inequality. Under lower ellipticity conditions on the nonlinear operator F, they were derived in [7,39], the former covering even many examples of quasi-linear equations. Descriptions via control theoretic and probabilistic arguments of the propagation sets of maxima (and minima) for Hamilton-Jacobi-Bellman equations were provided in [8,9]. Later results for fully nonlinear uniformly elliptic equations with linear gradient growth can be found in [36,Thm 5.1], see also [22] for other maximum principles for nonlinear elliptic operators. Property (SMP) has been also obtained in [2,Thm 5.1] via the weak Harnack inequality for some fully nonlinear PDEs (not necessarily uniformly elliptic) with superlinear and at most quadratic growth in the first order term.
Finally, recent contributions for fully nonlinear PDEs over Hörmander vector fields have been given in [10]. More recent progresses have been made on Hessian operators (mostly for truncated Laplacians), and they can be found in [16,17,28], while a rather different approach involving fully nonlinear uniformly elliptic PDEs and quasi-linear equations perturbed by suitable lower order terms has been proposed in [27,44]. Additional results on maximum principles for fully nonlinear equations can be found in the recent works [25,58], see also the references therein.
The literature of the corresponding equations on Riemannian manifolds is poor, although (SMP) and (SmP) are of local nature and one does not usually expect their failure depending on the properties of the considered manifold. For linear elliptic problems, the first result appeared in [23], see also [13,Prop 6.84] and [48,Appx A] for weak solutions to the manifold Laplace equation. The case of nonlinear equations modeled on the p-Laplacian perturbed by zeroth-order terms has been addressed in [53,Sect. 8.5]. A more general treatment of quasi-linear equations, involving even mean-curvature operators, can be found in [14,Sect. 4.2] under the name of finite maximum principle and in [1,Thm 3.10] for PDEs perturbed by first-order terms, see also [49] and the references therein. In the context of fully nonlinear operators, the only available results we are aware of in the context of Riemannian manifolds are those established by Harvey and Lawson [35], that however, cover the case of Hessian-type equations only.
Our approach is inspired by the landmark papers [7][8][9] and borrows several (viscosity) techniques from these works. In particular, we stress that the procedure implemented here is structured in the spirit of that used by Calabi [23] and the nonlinear one by Bardi and Da Lio [7], where solutions are meant in the weak viscosity sense, unlike the distributional framework developed in [3,53]. Our main result (SMP) that we prove in Theorem 4.2 for (1.1) is based on a classical barrier-type argument for viscosity solutions and works for those operators F that are elliptic according to the following notion and fulfilling a scaling condition like F x, cs, cq, cQ) ≥ η(c)F x, s, q, Q) for any s ∈ [−1, 0] and c ∈ (0, 1] , for some function η > 0 (see Sect. 3.1 for detailed assumptions). In particular, we highlight that the ellipticity condition reduces to the non-degeneracy property identified by Bardi and Da Lio [7] in the Euclidean setting whenever the manifold has nonnegative sectional curvature (see Remark 4.3). In Sects. 3.2 and 4 we list several examples to which our main Theorem 4.2 applies, among which equations driven by Pucci's, p-Laplace Beltrami, ∞-Laplacian and mean curvature operators, together with various singular nonlinear operators modeled on the p-and ∞-Laplacian for which the viscosity theory on Riemannian manifolds has been developed, see e.g. [41]. We will also provide a weak version of the Hopf boundary Lemma for viscosity solutions in Theorem 4.5. Furthermore, symmetric ellipticity and scaling conditions lead to a version of the (SmP), that is stated in Theorem 4.7 for reader's convenience.
Let us now pass to discuss property (SCP), for which the literature is less wide. For classical solutions to linear equations it is a straightforward byproduct of the (SMP). In the case of nonlinear problems, even degenerate, some additional conditions on the operator are needed. Property (SCP) for smooth solutions to quasi-linear equations can be found in [53,Thm 2.2.2] under the name of tangency principle. In the fully nonlinear case, strong comparison principles have been addressed in [7,Rem 3] (when one of the functions is C 2 via the arguments in [39]), by Ishii-Yoshimura in [36,Thm 5.3] for second order uniformly elliptic equations with Lipschitz growth in (u, du), N.S. Trudinger [56] for Lipschitz continuous viscosity solutions, Y. Giga and M. Onhuma [30] for semicontinuous solutions, see also [46,Thm 3.1] for other results on uniformly elliptic operators and [45] for the case of mean curvature equations, all of them in the Euclidean setting. More recently, strong comparison principles have been proved in [10] for degenerate Hamilton-Jacobi-Bellman equations, covering also problems structured over Hörmander vector fields, see also [42] for tangency principles results for fully nonlinear problems arising in conformal geometry.
In Sect. 5, we first set up a simple proof for Pucci's extremal operators when one of the involved functions is smooth (see Lemma 5.1) and then we prove (SCP) following a strategy implemented in [36] based on a combination of the (SMP) with the weak comparison principle in [4] that yields the result in the uniformly elliptic case. In particular, we assume F intrinsically uniformly continuous, uniformly elliptic in the sense that for any x ∈ M, (s, q, Q) ∈ J 2 x M, P ∈ Sym 2 (T x M) with P ≥ 0 and ellipticity constants 0 < λ ≤ , and satisfying x M, and prove (SCP) whenever the manifold has nonnegative sectional curvature, see Remark 5.7 for further comments on this restriction. This latter result is contained in Theorem 5.4 and essentially relies on proving that the difference w = u − v is a subsolution to a sort of linearized equation, following the path of [36].
As far as the weak comparison principle is concerned, the literature is huge when F is strictly proper since the results encompass even first order problems [5,26]. In the more general case of Riemannian manifolds, the only available contributions we are aware of in the second-order case have been obtained in [4]. In particular, when the manifold has nonnegative sectional curvature, weak comparison principles are obtained via the Riemannian counterpart of the Euclidean theorem on sums [26], see [4]. However, when (M, g) has negative curvature, the weak comparison continues to hold under a further uniform continuity assumption (see [4, cond (2b) on Cor 4.10] and Remark 5.7 below). We refer also to [43, Appx A] for further comparison principles on Riemannian manifolds for quasi-linear problems.
Under the mere properness of the operator one needs some form of ellipticity and the minimal conditions seem to be an open problem, see [12,37,39,40] and the more recent [10,11,25].
We finally mention that other kind of maximum principles (at infinity) on Riemannian manifolds can be found in the monographs [33,49], while Alexandroff-Bakelman-Pucci estimates together with maximum principles for subsolutions of linear problems were obtained in [20].
Plan of the paper Section 2 is devoted to some preliminaries on Differential Geometry. Section 3 introduces fully nonlinear second order equations on Riemannian manifolds together with the ellipticity definitions used throughout the paper. Section 4 comprehends the proof of the (SMP) and (SmP) with a list of prototype examples, while Sect. 5 applies the results to deduce (SCP).

Preliminaries of Riemannian Geometry
Let (M, g) be a C 3 -Riemannian n-manifold, i.e. a connected smooth manifold M of dimension n endowed with a symmetric (0, 2)-tensor field g ∈ C 3 (M, Sym 2 (T * M)), which induces at each point x ∈ M a positive definite inner product g x on the tangent space T x M. We denote by d the exterior derivative on M and by D the Levi-Civita covariant derivative on (M, g). We call domain any connected open subset ⊂ M. If the closure ⊂ M is compact, we say that has compact closure.
From now on, all the manifolds are assumed to be connected, all the Riemannian metrics are assumed to be of class C 3 and they are not necessarily complete.
The Riemannian manifold (M, g) is in particular a separable, locally compact length space by means of the induced Riemannian distance dist : M × M → R (see e.g. [47,Sect. 2]). We will denote by B(x, r ) the open metric ball centered at x ∈ M of radius r > 0 in M, by S(x, r ) := ∂ B(x, r ) its boundary and by B(x, r ) := B(x, r ) its closure. Notice that, by the classical Hopf-Rinow Theorem, if (M, g) is complete, then a domain has compact closure if and only if it is bounded (see e.g. [19,Thm 2.5.28]). Moreover, for any x ∈ M, we denote by Exp x the Riemannian exponential map at x and by inj(x) the injectivity radius of (M, g) at x.
Let x o ∈ M be a point. We denote by | · | the norm induced by g on the tangent bundle T M and by We recall that the injectivity radius x → inj(x) is continuous whenever g is complete (see e.g. [55]). Moreover, for any 0 is an embedded (n−1)-submanifold of class C 2 and, for any . For the sake of notation, given x, y ∈ M such that dist(x, y) < min{inj(x), inj(y)}, we denote by L xy : T x M → T y M the parallel transport along the unique minimizing geodesic from x to y.
Let now x o ∈ M and 0 < r < inj(x o ). We recall that the ball B(x o , r ) is said to be strongly geodesically convex if, for any ball B(y, r ) ⊂ B(x o , r ), any two points in B(y, r ) are joined by a unique minimizing geodesic which is entirely contained in B(y, r ). In this case, it holds that ( is called convexity radius of (M, g) at x o . We recall that the function x → R(x) is positive and, from the very definition, it is also 1-Lipschitz (see e.g. [59]). More precisely: Notice that in the Euclidean case, i.e. when (M, g) = (R n , , ), for any x o ∈ R n it holds that and any parallel transport L xy : R n T x R n → T y R n R n coincides with the identity map.

Notation
Let (M, g) be a Riemannian manifold and ⊂ M a domain. We denote by J 2 M → M the 2-jet bundle on M, which splits as by means of the Riemannian metric g. For any function ϕ ∈ C 2 ( , R), its 2-jet at x is given by We define the subbundle J 2 M ⊂ J 2 M by setting Let us consider a function F : J 2 (M) → R, which is assumed to satisfy the following fundamental condition: For the sake of shortness, for any function ϕ ∈ C 2 ( , R) we define Moreover, we set  (3.1) if it is both a viscosity subsolution and a viscosity supersolution.

Remark 3.2
Notice that, since we allow F to be singular at q = 0, this definition is slightly weaker than the one in [26,Sect. 2]. Other refined notions of viscosity inequalities via the envelopes have appeared when dealing with singular equations (we refer e.g. to [29,Def 2.3.1], [41] and references therein). However, we stress that if F is continuous and u : → R is of class C 2 with du = 0, then u is a viscosity subsolution (resp. viscosity supersolution) to F[u] = 0 if and only if it is a classical subsolution (resp. classical supersolution) to F[u] = 0.
We introduce now two notions of ellipticity which will be of central importance in this work (see also [7,Sec 1]). These are: Here, notice that R(x o ) and f x o have been defined in (2.2) and (2.1), respectively. For the sake of shortness, we say that F is partially elliptic if it is both lower and upper partially elliptic. Furthermore, we recall that F is uniformly elliptic if there exist two constants 0 < λ ≤ such that Notice that uniform ellipticity implies partial ellipticity (see Lemma 3.4), but the converse assertions do not hold true.
We also introduce two scaling properties which will play a role (see again [7, Sec 1]). These are: (l.s.p.) F has the lower scaling property if for any x o ∈ M there exist two functions Finally, we recall that F is said to be positively h-homogeneous, Clearly, positive h-homogeneity implies both the lower and the upper scaling properties, but the converse assertions do not hold true.

Universal Riemannian Operators
We recall here the construction of a distinguished kind of PDEs on Riemannian manifolds. More specifically, these have constant coefficients and are obtained from some Euclidean operators via the action of the orthogonal group O(n) (see [34,Sect. 5]).
Let (M, g) be a Riemannian manifold. Consider a function Then, one can construct an associated operator F : J 2 M → R in the following way. First, for any x ∈ M and for any orthonormal frame e = (e 1 , . . ., e n ) for T x M, we consider the linear isomorphism (iii) F has the lower scaling property if there exists a function η : The most famous example of universal Riemannian operator is the Laplace-Beltrami operator, which is associated to the Euclidean operator It is proper, positively 1-homogeneous, Lipschitz continuous and uniformly elliptic with ellipticity constants λ = = 1.

Remark 3.3
Another well-known example is the Monge-Ampère operator, which is associated to However, being the determinant not partially elliptic, it does not fall into the treatment of our analysis.
We are going to list below some other examples of universal Riemannian operators. From now on, for any A ∈ Sym(n), we will denote by μ 1 (A) ≤ · · · ≤ μ n (A) its ordered eigenvalues.

Pucci's Extremal Operators
Fix two constants 0 < λ ≤ and consider the set B λ, := B ∈ Sym(n) : λI ≤ B ≤ I . Then, we define M ± λ, : Sym(n) → R as The associated universal Riemannian operators M ± λ, are the so called Pucci's extremal operators, which are the prototype Hamilton-Jacobi-Bellman operators and, perhaps, the simplest example of Isaacs operators. They are proper, positively 1-homogeneous, Lipschitz continuous and uniformly elliptic with ellipticity constants 0 < λ ≤ . Moreover, it is well known that the uniform ellipticity of F can be characterized by means of the extremal operators M ± λ, via we get the following Proof Assume that F is uniformly elliptic with ellipticity constants 0 < λ ≤ and fix x o ∈ M. Then, for any and therefore F is lower partially elliptic by taking α large enough. On the other hand, the operator is uniformly elliptic as well. Hence, by means of our previous computation, we obtain that F − is lower partially elliptic, which is equivalent to say that F is upper partially elliptic.

Pucci's Operators from [51]
Fix a constant 0 < α ≤ 1 n and consider the setB α := B ∈ Sym(n) : B ≥ α I n , Tr(B) = 1 . Then, we define the functions We call the associated universal Riemannian operators P ± α original Pucci's operators, as they were firstly introduced in the Euclidean case by Pucci in [51]. They are proper, positively 1-homogeneous, Lipschitz continuous and uniformly elliptic with ellipticity constants 0 < α ≤ 1 − (n − 1)α, but do not allow to characterize uniform ellipticity of fully nonlinear operators as their extremal counterpart M ± λ, . Then, by means of (3.2), they are related with M ± λ, via

p-Laplacian Operator
Let 1 < p < ∞ and consider the function The associated universal Riemannian operator F p is the so called p-Laplacian, which normally appears in divergence form as It is proper, continuous and positively ( p−1)-homogeneous. Moreover, for any v = 0 we have and so F p is partially elliptic, even though it is not uniformly elliptic. We conclude by saying that the 1-homogeneous version of the p-Laplacian F p , called game-theoretic p-Laplacian, that is the operator F G p associated to is proper, continuous and positively 1-homogeneous. Moreover, it is easy to check that it is partially elliptic via condition (ii) in Sect. 3.1.

∞-Laplacian Operator
Consider the function The associated universal Riemannian operators F ∞ is the so called ∞-Laplacian, which is proper, continuous and positively 3-homogeneous. Since we conclude that F ∞ is partially elliptic. Similarly, the positively h-homogeneous version F h ∞ of the ∞-Laplacian, which is the operator associated to is proper, continuous and it is easy to check that it is partially elliptic.

Mean Curvature Operator
Consider the function The associated universal Riemannian operator F mc is the so called mean curvature operator, which can be written in divergence form as It is proper and continuous. Moreover, since it follows that F mc is partially elliptic.

The Strong Maximum Principle
Let (M, g) be a Riemannian manifold. First, we are going to prove a technical result which asserts the propagation of maxima on small balls for subsolutions to lower partially elliptic operators on (M, g). Namely Proof Assume by contradiction that there exist a finite radius 0 < r o < 2

as in (2.1) and consider the function
Hence, by the lower partial ellipticity and the lower scaling property, if we choose α > 0 big enough, we get 0 < c < 1 and We claim that ψ(x) ≤ 0 for any x ∈ U ⊂ B(x o , r ). In fact, assuming by contradiction that max x∈U ψ(x) > 0, then there exists x ∈ U such that ψ(x ) = max x∈U ψ(x) > 0.
But then, since u(x o ) + ε o h is of class C 2 , dh(x ) = 0 and u(x o ) ≥ 0, owing also to the fact that F is proper, we get which contradicts our previous claim. By construction, this implies that ψ ≤ 0 in B(x o , r ). Since ψ(x o ) = 0, together with the fact that u is a viscosity subsolution, we get a contradiction with (4.1).
As a direct corollary, we obtain the following version of the Strong Maximum Principle ((SMP) for short).
Indeed, under the hypothesis of nonnegative sectional curvature, by the Rauch Comparison Theorem it follows that ( [20,Lemma 3.1]). This is the same non-totally degeneracy condition found in the Euclidean case by Bardi-Da Lio [7].
We stress now that part of the proof of Proposition 4.1 can be adapted to prove a viscosity version of the Hopf boundary Lemma (see [24, Appx E] and [31,Lemma 3.4] for the corresponding statements in the linear case). Before doing that, we recall that: ⊂ M satisfies the interior ball condition at a boundary point x o ∈ ∂ if there exist y ∈ and 0 < r < R(y) such that B(y, r ) ⊂ and Under such hypothesis, for any point x ∈ B(y, r ) there exists a unique unit speed minimizing geodesic γ

2)
where γ x is the unique geodesic as above.
Proof Let B(y, r ) ⊂ be as in (i.b.c.). Let also α > 0, f y (x) = 1 2 dist(x, y) 2 as in (2.1) and consider the function By repeating the same argument as in the proof of Proposition 4.1, one can choose α > 0 big enough, 0 <r < R(y) − r small enough and 0 < ε < 1 small enough such that u(x) ≤ u(x o ) + εh(x) for any x ∈ U , with U : = B(x o ,r ) ∩ B(y, r ). Then, setting c := α exp −α r 2 2 we get lim inf which concludes the proof.

Remark 4.6
In the hypotheses of Theorem 4.5, if u ∈ C 1 (¯ ), then (4.2) reads as which is the classical statement of the Hopf Boundary Lemma (see [31,Lemma 3.4] and [24,Appx E]).
Finally, we remark that Theorem 4.2 can be used to derive a version of the Strong Minimum Principle ((SmP) for short). Indeed Since F − verifies all the hypothesis of Theorem 4.2, we get the thesis.
We are going to list below some examples that fulfill the hypotheses of our (SMP) and (SmP). Actually, we can cover more general universal Riemannian operators, e.g. we can consider Then, the operator F associated to F, which has been studied in [15], is proper and positively (β + 1)-homogeneous. Moreover, by [21, Lemma 2.10] we get and thus it is partially elliptic. Hence, both the (SMP) and the (SmP) hold true. This would extend [15,Proposition 2.15] to Riemannian manifolds.

Remark 4.10
To the best of our knowledge, this is the first (SMP) for viscosity solutions of the p-Laplace equation on Riemannian manifolds. Recently, some results have been established in [14] for C 1 distributional solutions of equations driven by the p-Laplacian on Riemannian manifolds. However, while in the Euclidean case weak (distributional) and viscosity solutions agree (see [38]), we are not aware of a similar result in the more general setting of Riemannian manifolds.

Example 4.11
We stress that neither (l.s.p.) nor (u.s.p.) hold true for the mean curvature operator F mc , but still it fulfills both (l.s.p.'), (u.s.p.') and so one can obtain the (SMP) and the (SmP) in virtue of Remark 4.8 (see also [7,Sec 3]). The same is true for the capillary surface equation Let us notice that the (SMP) for the mean curvature equation on Riemannian manifolds has been established recently in [14], while that for the capillary equation appears to be new. In the special case when F = − p , which is homogeneous of degree h = p − 1, the threshold k ≥ p − 1 has been found to be sharp in [57], cf also [7, Example 1] and the references therein.

Example 4.13
Let F be a Hessian operator, that is a universal Riemannian operator associated to an Euclidean operator F : Sym(n) → R. Assume that F is continuous, uniformly elliptic and positively 1-homogeneous. Let also b ∈ C 0 (M, T M) a continuous and bounded vector field on M. Then, the operators analyzed in [41] of the general form verify both the (SMP) and the (SmP).

An Application: The Strong Comparison Principle
Let (M, g) be a Riemannian manifold. In this section, we are going to establish some consequences of the Strong Maximum Principle.
In the case of linear equations, being the difference of two solutions a solution as well, it turns out that the (SCP) is equivalent to the (SMP). In the nonlinear case, it is worth observing that the (SCP) implies the (SMP) whenever a constant solves the equation. However, the converse may not hold. To have a guideline for the general case, we begin with the following   [6]).
Let us stress now that handling viscosity solutions requires more care and the use of the strong and weak maximum principles, as in [10,36]. Indeed, the fact that w = u −v is a viscosity subsolution to the initial equation is not straightforward in general (see e.g. [ for any x ∈ M and (s 1 , q 1 , Q), (s 2 , q 2 , Q) ∈ J 2 x M.

Remark 5.3
Any universal Riemannian operator F verifies (5.1) with ω = 0. Indeed, the O(n)-invariance implies that the corresponding Euclidean operator is of the form F(s, v, A) = f (s, |v|, eigenvalues ofA). Then, since |L * xy q| = |q| and Q, L * xy Q have the same eigenvalues, condition (i.u.c.) is fulfilled.
Then, the result reads as follows. and its lower semicontinuous envelope ( F ) * , which is defined by Then, we will need the following Notice that Lemma 5.5 can be obtained by following the proof of [36,Prop 5.5], which is stated in the Euclidean case (M, g) = (R n , , ). However, for convenience of the reader, we provide a proof of this result by following the authors' original approach.
Proof of Lemma 5.5 Arguing by contradiction, we assume that there exist a function ϕ ∈ C 2 ( , R) and a point Then, by hypothesis, it follows that: • v ϕ is a viscosity supersolution of F C [u] − (C + 1)u − 2c = 0 on ; • u is a viscosity subsolution of F C [u] − (C + 1)u = 0 on .
Since u ≤ v ϕ and u ∈ USC( ), v ϕ ∈ LSC( ), by using the Hahn-Katetov-Dowker Theorem (see [18,Thm 3.4.9]), combined with an approximation argument, it follows that there exists a Lipschitz continuous function h : B(x o , δ) → R such that (C + 1)u(x) ≤ h(x) ≤ (C + 1)v ϕ (x) + c for any x ∈ B(x o , δ). Therefore: • v ϕ is a viscosity supersolution of Hence, it follows that −w is a supersolution of G[u] = 0. Since G is continuous, uniformly elliptic and positively 1-homogeneous, the result is a consequence of Theorem 4.7.

Remark 5.6
The result holds true for equations driven by the normalized p-Laplacian on Riemannian manifolds under the assumptions of Theorem 5.4.

Remark 5.7
The assumption on the sectional curvature of (M, g) can be dropped at the expenses of assuming a further uniform continuity assumption in the variable x and D 2 u (see [4,Cor 4.10]). More precisely, Theorem 5.4 holds true in a general Riemannian manifold, without curvature conditions, if we replace the intrinsic uniform continuity (i.u.c.) by: (i.u.c.') for any ε > 0 there exists δ > 0 such that, for any x, y ∈ M with dist(x, y) < min{inj(x), inj(y)}, it holds dist(x, y) ≤ δ , −δ I ≤ P − L * xy Q ≤ δ I ⇒ F(y, s, q, Q) − F(x, s, L * xy q, P) ≤ ε (5 .2) for any (s, q, P) ∈ J 2 y M, Q ∈ Sym 2 (T * x M).
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