SCALAR CURVATURE DECREASE FROM A HYPERBOLIC METRIC

Abstract

We find an explicit $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on the 4-d hyperbolic space $\mathbb{H}^4$, for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ > 0 with the following property: $g_0$ is the hyperbolic metric on $\mathbb{H}^4$, 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.

1. INTRODUCTION

For any Riamannian manifold (Mk, 𝑔0), k ≥ 3 and a ball B ⊂ M, is there a C∞-continuous path of Riemannian metrics 𝑔t, 0 ≤ t ≤ ε on M such that the scalar curvatures of 𝑔t are strictly decreasing in t on B and that 𝑔t ≡ 𝑔0 on M＼B? This family, if exists, may be called a scalar curvature melting of 𝑔0 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 𝑔t, then the scalar curvatures satisfy

on B. As 𝑔t is deforming only inside a ball, it is more relevant to the linearization L𝑔 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 𝑔h, i.e. the metric with constant curvature -1. The derivative of the scalar curvature functional ds𝑔h (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 ℍ4 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 𝑔t. In this way we get a scalar curvature melting;

Theorem 1.1. There exists a C∞-continuous path of Riemannian metrics gt on ℍ4, for 0≤t ≤ ε for some number ε > 0 with the following property: 𝑔0 is the hyperbolic metric on ℍ4, the scalar curvatures of gt are strictly decreasing in t in an open ball and gt is isometric to g0 in the complement of the ball.

 

2. METRICS ON THE 4-D HYPERBOLIC SPACE

We start with a metric on ℝ4 of the form

where (r,𝜃); (𝜌,𝜎) are the polar coordinates for each summand of ℝ4 := ℝ2 × ℝ2 respectively, and f, h are smooth positive functions on ℝ4, 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 ℝ4. Then the hyperbolic metric corresponds to in the unit ball { (r, 𝜃, 𝜌, 𝜎)｜ r2 + 𝜌2 < 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 Fr=F′, Frr = F", and , hence the equation is F″+CF′+𝒟F = 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 c1(𝜌) = c2(𝜌) = 0 and we have a solution

Similarly we have

Hence

We choose α(r, 𝜌) = a(r)b(𝜌)(1 - r2 - 𝜌2)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:

Fig. 1.The graph of a.

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｜ F𝜌 = 0, Hr = 0}. By choosing a and b properly, 𝔗 becomes a thin subset in 𝒟.

One can check that the region 𝒟 lies within the 𝑔h-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.

 

3. A SCALAR-CURVATURE-DECREASING FAMILY

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.

 

4. DIFFUSION OF NEGATIVE SCALAR CURVATURE ONTO A BALL

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 Br(x) be the open ball of radius r with respect to centered at x. We may choose a point 𝜌 and a number 𝜖1 < 0:1 so that B2𝜖1(𝜌)⊂ 𝒟＼𝔗 as 𝔗 is a thin subset. Then on B𝜖1(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=𝜖1. 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 𝜙t 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 B𝜖1(p),, so choose m > 0 small so that 48me

In sum, we have and on B9+𝜖1(p and

We may have subtlety near the boundary ∂B9+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 ≤ t0 with some t0 > 0. Hence is strictly decreasing for 0 ≤ t ≤ t0 on . Setting , we get a scalar-curvature melting on B9+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.