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SCALAR CURVATURE DECREASE FROM A HYPERBOLIC METRIC
SCALAR CURVATURE DECREASE FROM A HYPERBOLIC METRIC
The Pure and Applied Mathematics. 2013. Nov, 20(4): 269-276
Copyright © 2013, Korean Society of Mathematical Education
  • Received : August 07, 2013
  • Accepted : November 14, 2013
  • Published : November 30, 2013
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About the Authors
YUTAE KANG
DEPARTMENT OF MATHEMATICS, SOGANG UNIVERSITY, SEOUL 121-742, REPUBLIC OF KOREAEmail address:lubo@sogang.ac.kr
JONGSU KIM
DEPARTMENT OF MATHEMATICS, SOGANG UNIVERSITY, SEOUL 121-742, REPUBLIC OF KOREAEmail address:jskim@sogang.ac.kr

Abstract
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.
Keywords
1. INTRODUCTION
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
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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
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(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)
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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 4 , for 0≤ t ≤ ε for some number ε > 0 with the following property : 𝑔 0 is the hyperbolic metric on 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 .
2. METRICS ON THE 4-D HYPERBOLIC SPACE
We start with a metric on ℝ 4 of the form
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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:
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where
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, etc..
Consider the unit ball centered at the origin in ℝ 4 . Then the hyperbolic metric corresponds to
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in the unit ball { ( r , 𝜃, 𝜌, 𝜎)| r 2 + 𝜌 2 < 1}. Note that
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in the rectangular coordinates. If we consider the deformation
where
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, the scalar curvature is given [2, p.59] by
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Substituting
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,
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,
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,
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and
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, we get;
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Put
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and
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Then
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We shall find F and H which satisfy
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and
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for some function α( r , 𝜌). For convenience we denote F r = F ′, F rr = F ",
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and
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, 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
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Choose u so that
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,
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Then
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Therefore the equation (2.1) becomes
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which is a well-known Euler-Cauchy equation. The general solution of this equation is
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Hence we have the solution
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Choosing c 1 (𝜌) = c 2 (𝜌) = 0 and
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we have a solution
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Similarly we have
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Hence
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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
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Note that this will make F ( r , 𝜌) = 0 and H ( r , 𝜌) = 0 when
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or
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.
A graph of a typical such function a (or b ) is given in the picture below:
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The graph of a.
Then
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and
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We set 𝒟 = {( r , 𝜃, 𝜌, 𝜙)| 0 ≤ r ,
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, 0 ≤ 𝜃, 𝜙 < 2𝜋}. Due to the conditions 1)-4) on a and b , the support of F and H lie in 𝒟. So,
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away from 𝒟 and from (2.2) its scalar curvature
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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
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Proposition 1 . There exist Riemannian metrics on
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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
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among the metrics in the previous section such that its scalar curvature
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is decreasing in 𝒟 \ 𝔗 and
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is hyperbolic in the complement of 𝒟.
We define a path of metrics:
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where
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and
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for the functions F and H as in (2.3). Then
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.
From (2.2) the scalar curvature is as follows;
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One can easily check
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and
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Note that inside 𝒟 the set of points with
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is identical to the set 𝔗. We see that
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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;
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for m , M > 0, t ≥ 0 defined by
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on
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and
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on
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. Also choose an
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with H = 0 on
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, H = 1 on
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and
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, for b > 0, 𝜖 > 0.
Let B r ( x ) be the open ball of radius r with respect to
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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
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on B 𝜖1 (𝜌) when 0 < t < c for some number c .
Let
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be
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, where ϱ( q ) is the
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-distance from 𝜌 to q
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and let
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be
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. We choose b = 9 and 𝜖=𝜖 1 . We consider the Riemannian metric
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, where
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Here m and M will be determined below. The scalar curvature is as follows;
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Setting
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, we have
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and
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As 𝜙 t is of second degree in t and B t=0 = -12, we readily get
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and
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As
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for a function f := f (ɐ), we compute
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Then we can readily see in (4.1) that
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for some large M > 0.
On B 𝜖1 ( p ),
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, so choose m > 0 small so that 48 me
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In sum, we have
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and
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on B 9+𝜖1 ( p and
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We may have subtlety near the boundary ∂ B 9 + ε1 ( p ), so we add the following argument.
On
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there exists
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such that
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is strictly decreasing for
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For a moment we set 𝜅 = 9 + ε1 - ɐ,
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and
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. On
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, so
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We have
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As M is large and m small,
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for 0 < t t 0 with some t 0 > 0. Hence
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is strictly decreasing for 0 ≤ t t 0 on
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. Setting
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, we get a scalar-curvature melting
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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.
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MOE) (No.NRF-2010-0011704).
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