We find an explicit
C
^{∞}
continuous path of Riemannian metrics
g
_{t}
on the 4d 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.
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
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 4dimensional 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
.
2. METRICS ON THE 4D HYPERBOLIC SPACE
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 wellknown EulerCauchy 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:
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｜
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.
3. A SCALARCURVATUREDECREASING 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
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 scalarcurvature 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.
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MOE) (No.NRF20100011704).
View Fulltext
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