Instantaneously complete Yamabe flow on hyperbolic space

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\geq3$. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.

We prove existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension m ≥ 3. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.
The Yamabe flow was introduced by Richard Hamilton [11]. It describes a family of Riemannian metrics g(t) subject to the equation ∂ t g = −Rg and tends to evolve a given initial metric towards a metric of vanishing scalar curvature. Hamilton showed that global solutions always exist on compact manifolds without boundary. Their asymptotic behaviour was subsequently analysed by Chow [5], Ye [19], Schwetlick and Struwe [15] and Brendle [3,4]. Less is known about the Yamabe flow on noncompact manifolds. Daskalopoulos and Sesum [6] analysed the profiles of self-similar solutions (Yamabe solitons). Ma and An [13] proved short-time existence of Yamabe flows on noncompact, locally conformally flat manifolds M under the assumption that the initial manifold (M, g 0 ) is complete with Ricci tensor bounded from below. More recently, Bahuaud and Vertman [1,2] constructed Yamabe flows starting from spaces with incomplete edge singularities such that the singular structure is preserved along the flow.
In dimension m = 2 the Yamabe flow coincides with the Ricci flow. Peter Topping and Gregor Giesen [16,7,17] introduced the notion of instantaneous completeness and obtained existence and uniqueness of instantaneously complete Ricci/Yamabe flows on arbitrary surfaces. The analysis of the flow on the hyperbolic disc plays an important role in their work. It relies on results which exploit the fact that the Ricci tensor is bounded by the scalar curvature in dimension 2.
The goal of this paper is to find techniques which allow a generalisation of Giesen and Topping's results to the Yamabe flow on hyperbolic space (H, g H ) of dimension m ≥ 3. (H, g H ) is a complete, noncompact, simply connected manifold of constant sectional curvature −1 and it is conformally equivalent to the Euclidean unit ball (B 1 , g E ).
Definition. A family (g(t)) t∈ [0,T ] of Riemannian metrics on a manifold M with scalar curvature R = R g(t) is called a Yamabe flow, if ∂ ∂t g = −R g. The family (g(t)) t∈[0,T ] is called instantaneously complete, if the Riemannian manifold (M, g(t)) is geodesically complete for every 0 < t ≤ T .
Since the Yamabe flow preserves the conformal class of the metric, any conformally hyperbolic Yamabe flow (g(t)) t∈[0,T ] on H is given by g(t) = u(·, t) g H , where the conformal factor u : H × [0, T ] → R is a positive function evolving by the equation where m = dim H, where ∆ g H denotes the Laplace-Beltrami operator with respect to the hyperbolic background metric g H and where |∇u| 2 g H = g H (∇u, ∇u). Introducing the exponent η := m−2 4 to define U = u η , equation (1) is equivalent to which follows by virtue of (η −1) = (m−6) 4 and 1 η ∆ g H u η = (η −1)u η−2 |∇u| 2 g H +u η−1 ∆ g H u. While equation (2) has a simpler structure, pointwise bounds on u follow easier from equation (1). We prove the following statements.
Theorem 1 (Existence). Let g 0 = u 0 g H be any (possibly incomplete) conformal metric on (H, g H ) with bounded conformal factor 0 < u 0 ∈ C 4,α (H) and scalar curvature R g 0 ≤ K 0 bounded from above. Then, for any T > 0 there exists an instantaneously complete family of metrics (g(t)) t∈[0,T ] satisfying the Yamabe flow equation in H × [0, T ], Moreover, g(t) ≥ m(m − 1) t g H for any t ∈ ]0, T ]. As t ց 0, the metric g(t) converges locally in class C 2 to g 0 .
Remark. On noncompact, locally conformally flat manifolds M, Ma and An [13] require bounded scalar curvature, a lower bound on the Ricci tensor and completeness of (M, g 0 ) for short-time existence and additionally non-positive scalar curvature for global existence.

Remark.
In the Poincaré ball model for H, the conformally hyperbolic initial metric g(0) can be compared to the Euclidean metric, whose pullback we also denote as g E . Assumption (i) means, that the initial manifold (H,g(0)) is incomplete and has finite diameter.
Assumption (ii) implies instantaneous completeness of g(t). We conjecture that instantaneously complete, conformally hyperbolic Yamabe flows always satisfy (ii). For rotationally symmetric flows, this is proved in Proposition 2.2.
The instantaneously complete flow Topping [16] constructs on 2-dimensional manifolds has a certain maximality property which we also observe in higher dimensions: Theorem 2 implies, that if g 0 ≤ b g E , then the Yamabe flow constructed in Theorem 1 is maximally stretched in the sense that any other Yamabe flow with the same or lower initial data stays below it.
Remark. Theorem 3 shows that the results about instantaneously complete Yamabe flow on hyperbolic space do not equally hold on arbitrary manifolds of dimension m ≥ 3. It contrasts with the 2-dimensional case, where instantaneously complete Yamabe flows always exist [7].
For example, there does not exist an instantaneously complete Yamabe flow starting from the punctured unit sphere (Ṡ m , g S m ) in dimension m ≥ 3. Indeed, if π :Ṡ m → R m is stereographic projection, then π * g S m = 4(1 + |x| 2 ) −2 g R m and Theorem 3 applies.
Acknowledgements. The author is very grateful to the anonymous referee for his work and care including valuable comments and suggestions.

Existence
In this section, we prove Theorem 1. As a first step, short-time existence of a solution u to equation (1)  In a second step we derive uniform gradient estimates, which do not depend on the domain. By considering an exhaustion of H with convex, bounded domains, we obtain a locally uniformly bounded sequence of solutions which allows a subsequence converging to a solution of (1) on all of H.
For small times t > 0, we expect the solution u to (3) to be close to the solutionũ of the linear problem Since Ω is bounded and since u 0 > 0 in H, there exists some δ > 0 depending on Ω and u 0 such that u 0 ≥ δ in Ω. Therefore, equation (5) is uniformly parabolic with regular coefficients and the compatibility conditions given in (4) are satisfied. According to linear parabolic theory [12, § IV.5, Theorem 5.2], problem (5) has a unique solutioñ In particular,ũ ≥ ε on Ω × [0, T ] for some ε > 0 depending on Ω andũ. Proof. A solution u to (3) is of the form u =ũ + v, whereũ solves (5) and Given the Hölder exponent 0 < α < 1, let . then is well-defined because the compatibility conditions (4) imply that at every p ∈ ∂Ω for every v ∈ X, we have The linearisation of Q[ũ] aroundũ ∈ C 2,α;1, α 2 (Ω × [0, T ]) defines the linear operator The map S is Gateaux differentiable at 0 ∈ X with derivative The mapping u → L(u) is continuous nearũ becauseũ is bounded away from zero. Hence, DS(0) is in fact the Fréchet-derivative of S at 0 ∈ X. Moreover, the linear operator ∂ ∂t − L(ũ) is uniformly parabolic. Let f ∈ Y be arbitrary. By definition, 0 = f (·, 0) is satisfied on ∂Ω which is the first order compatibility condition for the linear parabolic problem As before, linear parabolic theory states that (7) has a unique solution w ∈ X. Therefore, the linear map DS(0) : X → Y is invertible.
By the Inverse Function Theorem (Proposition A.1), S is invertible in some neighbourhood V ⊂ Y of S(0). We claim that V contains an element e such that e(·, t) = 0 for 0 ≤ t ≤ ε and sufficiently small ε > 0. Let We claim θf ∈ V for sufficiently small ε > 0. Sinceũ is smooth in Ω × [0, T ], we have f ∈ C 1 (Ω × [0, T ]). Since at t = 0, we have we can estimate Let t, s ∈ [0, T ] such that t > s. If s > 2ε, then (f − θf )(·, s) = (f − θf )(·, t) = 0. Therefore, we may assume s ≤ 2ε. In this case we estimate Due to (8), the special case s = 0 reduces to Since the left-hand side of (11) vanishes for t > 2ε, we have in fact If |t − s| < ε, estimate (10) directly implies If |t − s| ≥ ε, we replace the estimate by Therefore, For the spatial Hölder seminorm, we obtain a similar estimate from (9) and where d(x, y) denotes the Riemannian distance between x and y in (H, g H ) and where convexity of Ω is used. To conclude, f − θf Y ≤ Cε β−α f C 1 . Thus, θf belongs to the neighbourhood V of f if ε > 0 is sufficiently small. By construction, S −1 (θf ) is a solution to (6) in Ω × [0, ε]. Redefining T = ε > 0, we obtain the claim.

Local estimates
Let Ω ⊂ H be a smooth, convex, bounded domain. Let u ∈ C 2,α;1, α 2 (Ω × [0, T ]) be a solution to the nonlinear problem (3) as determined in Lemma 1.1. Restricting the hyperbolic background metric g H to Ω, we obtain the Yamabe flow g(t) = u(·, t)g H on Ω with initial metric g 0 = u 0 g H . In order to estimate the scalar curvature R = R g(t) of (Ω, g(t)) by means of the maximum principle, we will assume u 0 ∈ C 4,α (Ω) such that R g 0 ∈ C 2,α (Ω) and specify the parabolic boundary data φ explicitly. We define which is the relative initial velocity of the Yamabe flow in question compared to the "big bang"-Yamabe flow m(m − 1)tg H . Defining the constant we have |v| ≤ 2u 0 κ. For s ≥ 0, let 1] denotes the characteristic function of the interval [0, 1]. Figure 1: Graph of the function ψ.
In Ω × [0, T ] we can express the scalar curvature in the form where the right hand side satisfies the lower bound (16) and the upper bound (18). Scalar curvature evolves by the equation (see [ Let Let ε > 0 be sufficiently small depending only on u 0 , Ω and m, such that (16) holds and such that additionally, R g 0 ≥ − 1 ε in Ω. In the argument above we replace w(t) by − 1 t+ε and conclude R ≥ − 1 t+ε analogously. Lemma 1.3 (Upper and lower bound). Let 0 < u ∈ C 2,α;1, α 2 (Ω × [0, T ]) be a solution to problem (3) with boundary data (13) and bounded initial data u 0 > 0. Then, for Proof. From the equation for u, we deduce that given any constant c ∈ R the function Since (14) and the parabolic maximum principle (Proposition A.2) implies w ≥ 0 (respectively Proof. With derivatives and inner products taken with respect to g H , we have by (1) are uniformly bounded away from zero and from above and |∇u|, |∇v|, ∆v are bounded functions in Ω.
Lemmata 1.2 and 1.3 yield uniform bounds on the function U and the scalar curvature R in Ω × [0, T ]. Therefore, equation (21) implies In particular, since U = u η is bounded away from zero by Lemma 1.3, we have Hence, the equation has sufficiently regular coefficients for linear parabolic theory [12, § IV.5, Theorem 5.2] to apply: It follows that V = U is the unique solution to (22) with the given initial and boundary data. Moreover, U satisfies Since U is bounded away from zero in Ω × [0, T ], the claim follows.
Proof. According to Lemma 1.1, problem (3)  , that for some Ω the maximal existence time is T * < 1 K 0 . Then, Lemma 1.5 implies that u can be extended to u ∈ C 2,α;1, α 2 (Ω × [0, T * ]) and that u(·, T * ) ∈ C 2,α (Ω) is suitable initial data for problem (3). The boundary data (13) are defined also for t ≥ T * and they are compatible with u(·, T * ) at time T * . Therefore, we may apply Lemma 1.1 to extend the solution regularly in time in contradiction to the maximality of T * .

Uniform estimates
We assume that the initial metric g 0 = u 0 g H and its scalar curvature satisfy the upper bounds u 0 ≤ C 0 and R g 0 ≤ K 0 in H with some constant K 0 ≥ 0. Let 0 < T < 1 K 0 be fixed. From the previous section we recall that for any smooth, bounded domain Ω ⊂ H, there exists a uniformly bounded solution u of (3) on Ω × [0, T ]. However, the previous Hölder estimates on u may depend on the domain Ω. In the following, we derive independent bounds. As before, spatial derivatives and inner products are taken with respect to the hyperbolic background metric but in the following we will suppress the index g H to ease notation.
where the constant C depends on the dimension m and the constants C 0 , K 0 , T but not on ℓ. Similar bounds hold for higher derivatives of U.
Together with Ric g H = −(m − 1)g H , we apply (24) in the following computation. Hence, We insert (25) into (23) and resubstitute w = U p |∇U| 2 to obtain Choosing p = − 1 2 and deducing 2 m−1 R ≤ K 1 from Lemma 1.2 for some constant K 1 ≥ 0 depending on K 0 and T , we obtain and such that χ −3 |∇χ| 4 ≤ c 2 m which will be used later. (27) and (28) lead to Young's inequality for a, b ∈ R, δ > 0 and p, q > 1 with 1 p + 1 q = 1 states that We apply it with p = 4 3 and q = 4 and recall w = U − 1 2 |∇U| 2 , i.e. |∇U| = w Young's inequality with p = 2 = q also yields Let the sum of all terms in (29) containing ∆(χw) or ∇(χw) be denoted by and let the largest of the occurring factors which depend only on the dimension m be denoted by C m . Then Choosing δ = 1 16 , we have −( 1 4 − 2δ) = − 1 8 < 0. Since −χw 2 ≤ −(χw) 2 and since for any c 1 , c 2 > 0, we obtain with a different constantC m . By Lemma 1.3, we have the right hand side of (30) is bounded from above by a spatially constant, positive function f ∈ L 1 ([0, T ]). Let F ′ (t) = f (t) with F (0) = max(χw)(0) define a primitive function F for t → f (t). Then, we finally have Since the Yamabe flow equation is only of second order, similar estimates on higher derivatives of U follow analogously.
Proof of Theorem 1. Let η = m−2 4 and let the initial metric g 0 = u 0 g H be given by 0 < U 0 = u η 0 ∈ C 2,α (H). Let B r be the metric ball of radius r > 0 around the origin in (H, g H ). Then, B 1 ⊂ B 2 ⊂ . . . ⊂ H is an exhaustion of H with smooth, bounded domains. We fix 0 < T < 1 K 0 and choose φ as in (13). By Corollary 1.6, the problem is solvable for every k > 0. According to Lemma 1.7, the sequence {U k | B 1 ×[0,T ] } 2≤k∈N is uniformly bounded in C 2,α;1, α 2 (B 1 × [0, T ]). Since B 1 is a bounded domain, the embedding C 2,α;1, α 2 (B 1 × [0, T ]) ֒→ C 2;1 (B 1 × [0, T ]) is compact and we obtain a subsequence Λ 1 ⊂ N such that to a solution of the Yamabe flow equation (2) on B 1 . We repeat this argument to obtain a subsequence Λ 2 ⊂ Λ 1 such that converges to a solution of (2) on B 2 . Iterating this procedure leads to a diagonal subsequence of {U k } 2≤k which converges to a limit U ∈ C 2;1 (H × [0, T ]) satisfying the Yamabe flow equation (2). Since the bounds from Lemma 1.3 are preserved in the limit, we have m(m − 1)t ≤ U It remains to show that the Yamabe flow constructed above can be extended in time. Let T * be the supremum over all T > 0 such that there exists a Yamabe flow g(t) = u(·, t)g H on H which is defined for t ∈ [0, T ] and satisfies u(·, 0) = u 0 as well as We have already shown T * ≥ 1 K 0 . Suppose, T * < ∞ and let 0 < ε < 1 5 T * be arbitrary. For T = T * − ε, there exists u : H × [0, T ] → R satisfying u(·, 0) = u 0 together with estimate (31) and equation (2) which can be written in divergence form: where we recall η = m−2 4 . Around an arbitrary point p ∈ H, we choose geodesic normal coordinates x and given 0 < r < 1 3 √ T we consider the parabolic cylinder According to (31) and the choices of r, ε and T , we have Hence, (32) can be interpreted as a linear equation with uniformly bounded coefficient 1 u . Therefore we may apply parabolic DeGiorgi-Nash-Moser Theory [12, Theorem III.10.1] (see also [18]) to equation (32) in order to obtain u η+1 for some 0 < α < 1 and some constants C, C ′ depending only on the indicated quantities. In particular, the Hölder estimate (33) holds uniformly in p ∈ H. As in the proof of Lemma 1.5 we obtain u η C 2,α;1, α 2 (Qr) ≤ C ′′ (m, T * , sup u 0 ). Consequently, the scalar curvature R = −(m − 1)u −η−1 ( 1 η ∆u η + mu η ) stays uniformly bounded up to time T by some constant K 1 (m, T * , sup u 0 ). With the same methods as before, we can extend the solution, first locally in bounded domains and then via exhaustion in all of H. As initial data, we choose u 1 = u(·, T −ε). This allows us build compatible boundary data from a suitable extension of u| H×[T −ε,T ] . It also ensures the extended solution to be regular by Lemma 1.4

. The extended solution is defined in
we obtain a contradiction to the maximality of T * .

Uniqueness
This section contains the proofs of Theorems 2 and 3. As before, (H, g H ) denotes hyperbolic space of dimension m ≥ 3 and g E = h −2 g H the pullback of the Euclidean metric to H, where h > 0 is a smooth function provided by the Poincaré ball model.

Upper and lower bounds
The Yamabe flow constructed in the previous section satisfies the upper and lower bounds given in (31). The aim of this section is to find conditions under which any Yamabe flow necessarily satisfies such bounds. Since f g H = b g E is a flat metric, it is a static solution to the Yamabe flow equation As ϕ ′ • r A is non-positive in H and identically zero in the unit ball around the origin, we may apply Lemma A.3 stating ∆r ≤ 2(m − 1) and Lemma A.4 about cutoff functions (both given in the appendix) to estimate

Equation (34) and estimate (35) then yield
Denoting v = u η − f η as before, we obtain lim sup τ ց0 The assumption g(0) ≤ b g E implies max(u η − f η )(0) ≤ 0. Since t 0 > 0 is arbitrary, we may apply Lemma A.6 stated in the appendix to conclude Letting A → ∞ such that φ → 1 pointwise on H, we obtain A similar approach as for Proposition 2.1 leads to a proof of Theorem 3.
Proof of Theorem 3. Suppose, g(t) = u(·, t)g R m is a Yamabe flow on R m for t ∈ [0, T ] with g(0) = g 0 . Then where η = m−2 4 and ∆ denotes the Euclidean Laplacian. In the proof of Proposition 2.1, we replace the equation for u η by (37) and f by f With a cutoff function φ as in (35), we gain which means that (R m , g(t)) is geodesically incomplete.  (a, t) . Then, whenever r > a, Proof. Given a conformal metric g = ug H on H and any smooth function f : H → R, In the case f = ̺(·, t) and since ∆ g H r = m−1 tanh r by Lemma A.3, we have where we used the assumption of rotational symmetry. If r > a, then Moreover, with For any b, c > 0 and all X ∈ R the inequality which are smooth in H × ]0, T ] because ϕ is constant around zero. We remark that while ϕ ′ denotes the actual first derivative of the function ϕ of one variable, χ ′ is just a convenient shorthand notation. In fact, |∇χ| 2 g = |χ ′ ∇̺| 2 g = |χ ′ | 2 and ∆ g χ = χ ′′ + χ ′ ∆ g ̺. Hence, Recall that χ ′ ≤ 0. With the estimate from Lemma 2.3, we have Surprisingly, since ∂ ∂r χ = χ ′ √ u, the term ∂χ ∂r ∂v η ∂r = 2 ∇χ, ∇v η g cancels the last term in (45). Thus, By Lemma A.4, we may choose ϕ such that there exists a constant C depending only on m such that With this choice, By Young's inequality, ab ≤ a p p + b q q for any a, b ≥ 0 and p, q > 1 with 1 p + 1 q = 0. We apply it with p = 2η + 2 to estimate and with p = η + 1 to obtain Consequently, introducing the constantC = C 2η+2 + (A + 1)C η+1 , Provided that 0 < ε < 1 4 , the term involving χv η+1 in (48) has a negative sign. In this case, since 0 ≤ χ ≤ 1, we may replace (48) by The assumption of instantaneous completeness of g(t) implies that ̺(r, t) → ∞ as r → ∞ for every t ∈ ]0, T ]. Therefore, χ = ϕ • (ε̺) is compactly supported in H for every t ∈ ]0, T ] and w : ]0, T ] → ]0, ∞[ given by is well-defined. Let t 0 ∈ ]0, T ] be arbitrary. Let q 0 ∈ H such that w(t 0 ) = χv η (q 0 , t 0 ). We compute which shows that w is decreasing as long as w η+1 η ≥C √ ε. Hence, Letting ε ց 0 such that χ → 1 pointwise in H proves the claim.

Generalisation of Topping's interior area estimate
Topping [17] proves uniqueness of instantaneously complete Ricci flows on surfaces by estimating differences of area. In the following, we adapt his method to the Yamabe flow in dimension m ≥ 3. The Poincaré ball model realises hyperbolic space (H, g H ) as the unit ball in R m equipped with polar coordinates P : ]0, 1[ × S m−1 → H mapping (ρ, ϑ) → ρϑ and Riemannian metric In this section however, logarithmic polar coordinatesP : ]0, ∞[ × S m−1 → H given byP (s, ϑ) = P (e −s , ϑ) are more suitable. We record and note that the Riemannian manifold (Z, ζ) Proof of Theorem 2. Let g(t) = u(·, t)g H andg(t) = v(·, t)g H be two Yamabe flows on H for t ∈ [0, T ]. Let U, V : Z × [0, T ] → ]0, ∞[ such thatP * g(t) = U(·, t) ζ and P * g (t) = V (·, t) ζ. From equation (2) follows that U and V both solve where 4η = (m − 2) = 1 m−1 R ζ and ∆ ζ = ∂ 2 ∂s 2 + ∆ g S m−1 is the Laplace-Beltrami operator with respect to the metric ζ = ds 2 + g S m−1 on Z. Note that U, V and their derivatives with respect to s have exponential decay for s → ∞. In fact, Since u is positive and regular at the origin, [u η−1 ∂u ∂ρ ](e −s θ, t) stays bounded as s → ∞.
By assumption (i) and equation (50), applying Proposition 2.1 tog(t), we have Combining (54) and (55), we obtain where x + := max{x, 0}. Abbreviating w := V η+1 − U η+1 , we have for every fixed t ∈ ]0, T ] and every 0 < τ < t We obtain ∂w ∂t (·, t) ϕ dµ ζ where we may interchange limit and integral because with f := V η − U η and (51) we have which is bounded in Z S × [0, T ] with exponential decay for s → ∞. We claim that Indeed, let (m k ) k∈N be a sequence of regular values for f (·, t) such that m k ց 0 as k → ∞. Then, {f (·, t) > m k } ⊂ Z is a regular, open set with outer unit normal ν in the direction of −∇f . Moreover, since f and ∇f have exponential decay for s → ∞ according to (52) and (53), since ϕ(S) = 0 and since ∇ϕ is supported in [S, s 0 ], we have by Green's formula Passing to the limit k → ∞ proves (58) since Z S \ Z s 0 is a bounded domain. Hence, Introducing the exponent λ ∈ ]0, 1 3 [ we modify estimate (56) as follows.
In the case that g(t) andg(t) both satisfy (ii) and g(0) =g(0) ≤ b g E , the reverse inequalityg(t) ≥ g(t) follows similarly by switching the roles of U and V .

A. Auxiliary results
The previous sections depend on some standard results and computations which we collect in this appendix for convenience of the reader. In the following proposition, we denote partial derivatives by subscripts and understand a sum m i=1 whenever an index i appears twice in an expression.
where the function c < λ ∈ R is bounded from above and ellipticity a ij ξ i ξ j ≥ 0 for all ξ ∈ R m holds uniformly. Then, u ≤ 0 in Ω × [0, T ].
Proof. The Poincaré ball model realises hyperbolic space (H, g H ) as the unit ball equipped with polar coordinates P : ]0, 1[ × S m−1 → H mapping (ρ, ϑ) → ρϑ and conformal Riemannian metric P * g H = h 2 g E , where Here, g S m−1 is the standard metric on the unit sphere S m−1 ⊂ R m and g E is the Euclidean metric on the unit ball in R m . We denote the radial coordinate on the unit ball by ρ and reserve r for the hyperbolic distance which is given by On the one hand, equation (38) implies Combined with (67), the claim follows.
Lemma A.4 (cutoff function). Let ε > 0 and a, b > 0 be real parameters. Then there exists a non-increasing cutoff function ϕ ∈ C 2 (R) given by which satisfies the inequality with a constant C depending only on a, b and ε.