We find an explicit C ^∞ -continuous path of Riemannian metrics g _t on the 4-d hyperbolic space ℍ ⁴ , for 0 ≤ t ≤ ε for some number ε > 0 with the following property: g ₀ is the hyperbolic metric on ℍ ⁴ , the scalar curvatures of g _t are strictly decreasing in t in an open ball and g _t is isometric to the hyperbolic metric in the complement of the ball. For any Riamannian manifold ( M ^k , 𝑔 ₀ ), k ≥ 3 and a ball B ⊂ M , is there a C ^∞ -continuous path of Riemannian metrics 𝑔 _t , 0 ≤ t ≤ ε on M such that the scalar curvatures of 𝑔 _t are strictly decreasing in t on B and that 𝑔 _t ≡ 𝑔 ₀ on M ＼ B ? This family, if exists, may be called a scalar curvature melting of 𝑔 ₀ in B . This question is actually a small step toward Lohkamp’s conjecture on ricci curvature version [6, Section 10] . If there is a scalar curvature melting 𝑔 _t , then the scalar curvatures satisfy on B . As 𝑔 _t is deforming only inside a ball, it is more relevant to the linearization L _𝑔 of the scalar curvature functional on the space of Riemannian metrics restricted to a domain. According to Corvino [3, Theorem 4] , a scalar curvature melting of 𝑔 seems to exist when the formal adjoint (as defined on the space of functions which are square integrable on each compact subset of B ) is injective. Although this injectivity condition holds for generic metrics by Theorem 6.1 and Theorem 7.4 in [1] , it is not easy to check which metrics satisfy this. In the previous works we have studied explicit scalar curvature meltings of Euclidean metrics and one positive Einstein metric [4 , 5] . In this article we study the hyperbolic metric 𝑔 _h , i.e. the metric with constant curvature -1. The derivative of the scalar curvature functional d s _𝑔h (defined on a whole manifold M ) is surjective, but we do not know whether the above (locally defined) is injective or not. In any case, a merit of our construction is that it is explicit and provides a large scale melting. We shall first construct a family of Riemannian metrics on the 4-dimensional hyperbolic space ℍ ⁴ whose scalar curvatures decrease on a precompact open subset and are hyperbolic away from it. Then by conformal change of the metrics, we spread the negativity inside the subset over to a larger ball. In the process, we find a natural choice of parameter t to get 𝑔 _t . In this way we get a scalar curvature melting; Theorem 1.1. There exists a C ^∞ - continuous path of Riemannian metrics g _t on ℍ ⁴ , for 0≤ t ≤ ε for some number ε > 0 with the following property : 𝑔 ₀ is the hyperbolic metric on ℍ ⁴ , the scalar curvatures of g _t are strictly decreasing in t in an open ball and g _t is isometric to g ₀ in the complement of the ball . We start with a metric on ℝ ⁴ of the form where ( r ,𝜃); (𝜌,𝜎) are the polar coordinates for each summand of ℝ ⁴ := ℝ ² × ℝ ² respectively, and f , h are smooth positive functions on ℝ ⁴ , which are functions of r and 𝜌 only. Then by a straightforward computation one gets the scalar curvature: where , etc.. Consider the unit ball centered at the origin in ℝ ⁴ . Then the hyperbolic metric corresponds to in the unit ball { ( r , 𝜃, 𝜌, 𝜎)｜ r ² + 𝜌 ² < 1}. Note that in the rectangular coordinates. If we consider the deformation where , the scalar curvature is given [2, p.59] by Substituting , , , and , we get; Put and Then We shall find F and H which satisfy and for some function α( r , 𝜌). For convenience we denote F _r = F ′, F _rr = F ", and , hence the equation is F ″+ C F ′+𝒟 F = α. If we assume the solution is of the form F ( r ,𝜌) = u ( r ,𝜌) v ( r ,𝜌), the equation becomes Choose u so that , Then Therefore the equation (2.1) becomes which is a well-known Euler-Cauchy equation. The general solution of this equation is Hence we have the solution Choosing c ₁ (𝜌) = c ₂ (𝜌) = 0 and we have a solution Similarly we have Hence We choose α( r , 𝜌) = a ( r ) b (𝜌)(1 - r ² - 𝜌 ² ) ³ where a ( r ) and b (𝜌) are smooth functions satisfying Note that this will make F ( r , 𝜌) = 0 and H ( r , 𝜌) = 0 when or . A graph of a typical such function a (or b ) is given in the picture below: Then and We set 𝒟 = {( r , 𝜃, 𝜌, 𝜙)｜ 0 ≤ r , , 0 ≤ 𝜃, 𝜙 < 2𝜋}. Due to the conditions 1)-4) on a and b , the support of F and H lie in 𝒟. So, away from 𝒟 and from (2.2) its scalar curvature inside 𝒟 except the subset 𝔗 := {( r , 𝜃, 𝜌, 𝜙)∈ 𝒟｜ F _𝜌 = 0, H _r = 0}. By choosing a and b properly, 𝔗 becomes a thin subset in 𝒟. One can check that the region 𝒟 lies within the 𝑔 _h- distance 4 from the origin Proposition 1 . There exist Riemannian metrics on such that their scalar curvatures are less than that of the hyperbolic metric on the subset 𝒟＼𝔗 and they are hyperbolic away from 𝒟. We are going to show that there is a C ^∞ -continuous path among the metrics in the previous section such that its scalar curvature is decreasing in 𝒟 ＼ 𝔗 and is hyperbolic in the complement of 𝒟. We define a path of metrics: where and for the functions F and H as in (2.3). Then . From (2.2) the scalar curvature is as follows; One can easily check and Note that inside 𝒟 the set of points with is identical to the set 𝔗. We see that is strictly decreasing only on 𝒟＼𝔗. In order to have the right decreasing property, we need to diffuse the negativity (of scalar curvature) onto a ball containing 𝒟＼𝔗. Our argument in this section follows those in [4, Section 4] and [5, Section 4] with just a few differences in estimation. We use the following functions; for m , M > 0, t ≥ 0 defined by on and on . Also choose an with H = 0 on , H = 1 on and , for b > 0, 𝜖 > 0. Let B _r ( x ) be the open ball of radius r with respect to centered at x . We may choose a point 𝜌 and a number 𝜖 ₁ < 0:1 so that B _2𝜖1 (𝜌)⊂ 𝒟＼𝔗 as 𝔗 is a thin subset. Then on B _𝜖1 (𝜌) when 0 < t < c for some number c . Let be , where ϱ( q ) is the -distance from 𝜌 to q ∈ and let be . We choose b = 9 and 𝜖=𝜖 ₁ . We consider the Riemannian metric , where Here m and M will be determined below. The scalar curvature is as follows; Setting , we have and As 𝜙 _t is of second degree in t and B ｜ _t=0 = -12, we readily get and As for a function f := f (ɐ), we compute Then we can readily see in (4.1) that for some large M > 0. On B _𝜖1 ( p ), , so choose m > 0 small so that 48 me In sum, we have and on B _9+𝜖1 ( p and We may have subtlety near the boundary ∂ B ₉ + _ε1 ( p ), so we add the following argument. On there exists such that is strictly decreasing for For a moment we set 𝜅 = 9 + _ε1 - ɐ, and . On , so We have As M is large and m small, for 0 < t ≤ t ₀ with some t ₀ > 0. Hence is strictly decreasing for 0 ≤ t ≤ t ₀ on . Setting , we get a scalar-curvature melting on B ₉ + _ε1 (𝜌) for 0 ≤ t ≤ ε. Theorem 1.1 is proved. Remark 1. The argument in this article may be applicable to some other metrics. A more generalization, including spherical metrics, will appear later.