Deformation of the Weighted Scalar Curvature
Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal ²ϕ-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particu...
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2023
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| Цитувати: | Deformation of the Weighted Scalar Curvature. Pak Tung Ho and Jinwoo Shin. SIGMA 19 (2023), 087, 15 pages |
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| author | Ho, Pak Tung Shin, Jinwoo |
| author_facet | Ho, Pak Tung Shin, Jinwoo |
| citation_txt | Deformation of the Weighted Scalar Curvature. Pak Tung Ho and Jinwoo Shin. SIGMA 19 (2023), 087, 15 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal ²ϕ-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particular, we define a weighted vacuum static space and study locally conformally flat weighted vacuum static spaces. We then prove some stability results of the weighted scalar curvature on flat spaces. Finally, we consider the prescribed weighted scalar curvature problem on closed smooth metric measure spaces.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 087, 15 pages
Deformation of the Weighted Scalar Curvature
Pak Tung HO a and Jinwoo SHIN b
a) Department of Mathematics, Tamkang University, Tamsui, New Taipei City 251301, Taiwan
E-mail: paktungho@yahoo.com.hk
b) Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea
E-mail: shinjin@kias.re.kr
Received December 07, 2022, in final form October 30, 2023; Published online November 04, 2023
https://doi.org/10.3842/SIGMA.2023.087
Abstract. Inspired by the work of Fischer–Marsden [Duke Math. J. 42 (1975), 519–547],
we study in this paper the deformation of the weighted scalar curvature. By studying the
kernel of the formal L2
ϕ-adjoint for the linearization of the weighted scalar curvature, we
prove several geometric results. In particular, we define a weighted vacuum static space,
and study locally conformally flat weighted vacuum static spaces. We then prove some
stability results of the weighted scalar curvature on flat spaces. Finally, we consider the
prescribed weighted scalar curvature problem on closed smooth metric measure spaces.
Key words: weighted scalar curvature; smooth metric measure space; vacuum static space
2020 Mathematics Subject Classification: 53C21; 53C23
1 Introduction
A smooth metric measure space is the tuple
(
M, g, e−ϕdVg,m
)
, where (M, g) is a smooth Rie-
mannian manifold, dVg is the volume form of g, e−ϕdVg is a smooth measure determined by
ϕ ∈ C∞(M), and m is a dimensional parameter with 0 ≤ m ≤ ∞. It was first introduced
by Bakry and Émery in [3]. Smooth metric measure spaces have recently attracted a lot of
attention in Riemannian geometry. For example, they play an important role in Perelman’s
approach to the Ricci flow [28]. The weighted scalar curvature of the smooth metric measure
space
(
M, g, e−ϕdVg,m
)
is defined as
Rmϕ = R+ 2∆ϕ− m+ 1
m
|∇ϕ|2, (1.1)
where R is the scalar curvature of g, ∆ and ∇ are the Laplacian and the gradient of g, respec-
tively. The geometric meaning of the weighted scalar curvature is as follows: Let (Fm, h) be the
flat m-torus. We regard
(
M, g, e−ϕdVg,m
)
as the base of the warped product(
M × Fm, g ⊕ e−
2ϕ
m h
)
. (1.2)
Then the weighted scalar curvature Rmϕ is the scalar curvature of the warped product (1.2).
In [15], Fischer and Marsden studied the deformation of the scalar curvature. In order to
study this problem, they studied the kernel of formal L2-adjoint for the linearization of scalar
curvature. More precisely, they considered the scalar curvature as a function on the space
of Riemannian metrics, i.e., the scalar curvature map g 7→ R(g). Then they computed the
linearization
Lg(h) :=
d
dt
∣∣∣∣
t=0
R(g + th)
mailto:paktungho@yahoo.com.hk
mailto:shinjin@kias.re.kr
https://doi.org/10.3842/SIGMA.2023.087
2 P.T. Ho and J. Shin
of the scalar curvature map and its formal L2-adjoint L∗
g. They showed that a closed Riemannian
manifold with kerL∗
g = {0} is linearization stable and hence any smooth function sufficiently
close to the scalar curvature of the background metric can be realized as the scalar curvature of
a nearby metric. A Riemannian manifold (M, g) with kerL∗
g ̸= {0} is also called vacuum static
space, which has been widely studied by many authors. See [11, 12, 13, 25, 29, 30] and the
references therein. In recent years there has been developed a general theory for “conformally
variational invariants” which gives sufficient conditions to perturb a given Riemannian manifold
such that a given scalar Riemannian invariant achieves a given function [8, 9]. Specializing this
to scalar curvature recovers the result of Fischer–Marsden, and the results are also applicable
to Q-curvature, σ2-curvature, and many more invariants.
Inspired by [15], we study in this paper the deformation of the weighted scalar curvature.
To do this, we regard the weighted scalar curvature Rmϕ as a function on the space of Riemannian
metrics and the space of smooth functions,
(g, ϕ) 7→ R(g, ϕ) = R+ 2∆ϕ− m+ 1
m
|∇ϕ|2.
Let DRg,ϕ be the linearization of this weighted scalar curvature map, and let DR∗
g,ϕ be the
formal L2-adjoint of DRg,ϕ in the weighted sense. In Section 3, we find the precise expression
of DRg,ϕ and DR∗
g,ϕ. Similar to the vacuum static space, the weighted vacuum static space is
defined to be the smooth metric measure space with kerDR∗
g,ϕ ̸= {0} (see Definition 3.2). By
analyzing the kernel of DR∗
g,ϕ, we prove several geometric results. In particular, we prove the
following classification result of closed weighted vacuum static spaces.
Proposition 1.1. Let
(
M, g, e−ϕdVg,m
)
be a closed weighted vacuum static space. Then
(
M, g,
e−ϕdVg,m
)
is either isometric to a Ricci-flat manifold with ϕ being constant, or the weighted
scalar curvature Rmϕ is a positive constant. In particular, any weighted vacuum static space has
nonnegative constant weighted scalar curvature.
In Section 4, we prove the following.
Theorem 1.2. Let
(
M, g, e−ϕdVg,m
)
be a connected weighted vacuum static space of dimension
n ≥ 3 (not necessarily compact). If
(
M, g, e−ϕdVg,m
)
is locally conformally flat in the weighted
sense, then around any regular point of f , the manifold (M, g) is locally a warped product with
(n− 1)-dimensional fibers of constant sectional curvature.
We say that
(
M, g, e−ϕdVg,m
)
is locally conformally flat in the weighted sense if for each
p ∈ M , there is a neighborhood U of p on which there is a conformal factor u for which(
e2ug, e(m+n)ue−ϕdVg
)
=
(
gflat,dVgflat
)
. In [25], Kobayashi classified the conformally flat vacuum
static spaces. Independently, Lafontaine [26] obtained a classification of closed conformally flat
vacuum static spaces.
Linearization stability of the scalar curvature on flat spaces was studied in [15], and the corre-
sponding stability for Q-curvature was studied in [27]. Inspired by these results, in Section 5, we
consider the linearization stability of the weighted scalar curvature on flat spaces. In particular,
we have the following.
Theorem 1.3. For n ≥ 3, let
(
M, g, e−ϕdVg,m
)
be a compact smooth metric measure space
with g being flat, ϕ being constant, and m being a positive integer. There is an ϵ > 0 such that
if (g, ϕ) has nonnegative weighted scalar curvature and is ϵ-close to
(
g, ϕ
)
in C2, then g is flat
and ϕ is constant.
In Section 5, we also consider the prescribed weighted scalar curvature problem on closed
smooth metric measure spaces. We have the following.
Deformation of the Weighted Scalar Curvature 3
Theorem 1.4. Let
(
M, g0, e
−ϕ0dVg0 ,m
)
be a closed smooth metric measure space with weighted
scalar curvature Rmϕ0, and let K be a non-constant smooth function on M . If there is a constant
c0 > 0 satisfying
c0minK < Rmϕ0 < c0maxK,
there is a smooth metric measure
(
M, g, e−ϕdVg,m
)
such that its weighted scalar curvature is K.
We remark that, on closed manifolds, the prescribed scalar curvature problem has been
studied by Kazdan and Warner in [18, 19, 20, 21], and on compact Riemannian manifolds with
boundary, the problem of prescribing the scalar curvature in M and the mean curvature on the
boundary ∂M was studied in [14, 17].
After this paper was finished, it was brought to our attention that many more relevant
weighted invariants are constructed through the recent work of Khaitan [22, 23]. In light of the re-
sults of the σ2-invariant and renormalized volume coefficients in closed Riemannian manifolds, we
expect that results similar to ours also hold for the invariant constructed by Khaitan (cf. [2, 31]).
2 Smooth metric measure space
We collect in this section some basic definitions and facts for smooth metric measure spaces
which will be needed in the rest of the paper. First, we have the following.
Definition 2.1. A smooth metric measure space is a four-tuple
(
M, g, e−ϕdVg,m
)
, where (M, g)
is an n-dimensional Riemannian manifold, ϕ is a smooth function in M , and m ∈ (0,∞) is
a dimensional parameter.
We remark that in this paper we consider only the case where m is a positive finite number.
When m = ∞, the weighted scalar curvature is sometimes called the P -scalar curvature and
has been studied in [1].
Definition 2.2. Let
(
M, g, e−ϕdVg,m
)
be a smooth metric measure space and let (V, hV ) and
(W,hW ) be vector bundles with inner product over M , and let ⟨·, ·⟩V and ⟨·, ·⟩W be the corre-
sponding inner products
⟨ζ1, ζ2⟩V =
∫
M
hV (ζ1, ζ2)e
−ϕdVg, ⟨ξ1, ξ2⟩W =
∫
M
hW (ξ1, ξ2)e
−ϕdVg
on sections ζi ∈ Γ(V ) and ξi ∈ Γ(W ) determined by the measure e−ϕdVg. The weighted di-
vergence divϕ : Γ(W ) → Γ(V ) of an operator D : Γ(V ) → Γ(W ) is the (negative of the) formal
adjoint of D with respect to the inner products ⟨·, ·⟩V and ⟨·, ·⟩W , i.e., for all ζ ∈ Γ(V ) and
ξ ∈ Γ(W ), at least one of which is compactly supported in M ,
⟨D(ζ), ξ⟩W = −⟨ζ,divϕξ⟩V .
The weighted Laplacian ∆ϕ : C
∞(M) → C∞(M) is the operator ∆ϕ = divϕd.
Lemma 2.3 ([5, Lemma 3.5]). Let
(
M, g, e−ϕdVg,m
)
be a smooth metric measure space. The
weighted divergence divϕ is related to the usual divergence divg by
divϕ = eϕ ◦ divg ◦ e−ϕ,
where eϕ and e−ϕ are regarded as multiplication operators. In particular, we have the formulas
divϕω = divgω − ι∇ϕω, ∆ϕw = ∆gw − ⟨∇ϕ,∇w⟩
for all ω ∈ ΛkT ∗M and all w ∈ C∞(M).
4 P.T. Ho and J. Shin
When M is closed, it is well known that∫
M
⟨∇f,X⟩e−ϕdVg = −
∫
M
fdivϕ(X)e−ϕdVg (2.1)
for any vector field X in M .
Definition 2.4. Let
(
M, g, e−ϕdVg,m
)
be a smooth metric measure space. The Bakry–Émery
Ricci tensor Rcmϕ is the symmetric (0, 2)-tensor
Rcmϕ := Rc + Hessgϕ− 1
m
dϕ⊗ dϕ,
where Rc is the Ricci tensor with respect to g, and Hessg is the Hessian of ϕ with respect to g.
The weighted scalar curvature Rmϕ is defined as
Rmϕ := R+ 2∆gϕ− m+ 1
m
|∇ϕ|2,
where R is the scalar curvature with respect to g.
It is important to note that the weighted scalar curvature is in general not the trace of the
Bakry–Émery Ricci tensor. Indeed, there holds (see [5, formula (4.1)])
Rmϕ = trRcmϕ +∆ϕϕ.
The following proposition was proved in [5, Proposition 4.2].
Lemma 2.5. There holds
divϕRc
m
ϕ =
1
2
dRmϕ − 1
m
∆ϕϕdϕ.
3 Deformation of the weighted Scalar curvature
Throughout this section, we will always assume
(
M, g, e−ϕdVg,m
)
is an n-dimensional closed
smooth metric measure space (n ≥ 3) unless otherwise stated. We study the weighted scalar
curvature map
(g, ϕ) 7→ R(g, ϕ) := R+ 2∆ϕ− m+ 1
m
|∇ϕ|2.
We first compute the linearization of this map and its adjoint. To this end, we let M be the
space of Riemannian metrics on M . The following formula is known.
Lemma 3.1. Let k be a nonnegative integer, and let l > n
p + 2. The map R as a map
R :
(
Ck+2,α ∩ M
)
× Ck+2,α → Ck,α, or R :
(
W l,p ∩ M
)
× W l,p → W l−2,p is smooth. The
linearization of Rmϕ is given by
DRg,ϕ(h, ψ) :=
d
dt
∣∣∣∣
t=0
Rmϕ+tψ(g + th)
= divϕdivϕh−
〈
h,Rcmϕ
〉
−∆ϕ(trgh) + 2
(
∆ϕψ − 1
m
⟨dϕ,dψ⟩
)
.
Proof. See, for example, [7, Proposition 5.1]. ■
Deformation of the Weighted Scalar Curvature 5
The formal L2
ϕ-adjoint DR∗
g,ϕ of DRg,ϕ is defined as∫
M
fDRg,ϕ(h, ψ)e
−ϕdVg =
∫
M
〈
DR∗
g,ϕ(f), (h, ψ)
〉
e−ϕdVg.
Using (2.1) and Lemma 3.1, we can compute DR∗
g,ϕ as follows:
DR∗
g,ϕ(f) =
(
−(∆ϕf)g +Hessgf − fRcmϕ , 2∆ϕf +
2
m
(
⟨df, dϕ⟩+ f∆ϕϕ
))
.
Note that the adjoint operator DR∗
g,ϕ is overdetermined-elliptic, i.e., it has injective symbol.
Indeed, the principal symbol of DR∗ is given by
σ(DR∗
g,ϕ)x(ϵ)f =
((
∥ϵ∥2g − ϵ⊗ ϵ
)
f,− 2
m
∥ϵ∥2f
)
.
Then one can easily see that it is injective for ϵ ̸= 0. Since DR∗
(g,ϕ) is overdetermined-elliptic,
the operator DR(g,ϕ)DR∗
(g,ϕ) is strictly elliptic. Thus the kernel of this operator is composed
of smooth fields (assuming that g and ϕ are smooth) by elliptic regularity, and on a closed
manifold, this kernel is the same as the kernel of the adjoint DR∗
(g,ϕ), via the identity∫
M
fDR(g,ϕ)DR∗
(g,ϕ)(f)e
−ϕdVg =
∫
M
∣∣DR∗
(g,ϕ)(f)
∣∣2e−ϕdVg.
A similar computation shows that the image of DR(g,ϕ) is L
2
ϕ-orthogonal to the kernel of DR∗
(g,ϕ).
Definition 3.2. We say that the smooth metric measure space (not necessarily compact)(
M, g, e−ϕdVg,m
)
is a weighted vacuum static space if there exists a smooth function f ̸≡ 0
such that
−(∆ϕf)g +Hessgf − fRcmϕ = 0, and ∆ϕf +
1
m
(
⟨df,dϕ⟩+ f∆ϕϕ
)
= 0. (3.1)
We say that
(
M, g, e−ϕdVg,m
)
is a nontrivial weighted vacuum static space if there exists f
satisfying (3.1) which is not constant.
Note that f satisfies (3.1) if and only if f lies in the kernel of DR∗
g,ϕ. Hence,
(
M, g, e−ϕdVg,m
)
is a weighted vacuum static space if and only if the kernel of DR∗
g,ϕ is nontrivial.
Taking the trace of the first equation in (3.1) yields
(1− n)∆ϕf + f∆ϕϕ+ ⟨∇f,∇ϕ⟩ − fRmϕ = 0.
Combining this with the second equation in (3.1), we have
∆ϕf = −
fRmϕ
n+m− 1
. (3.2)
So the first equation in (3.1) can be written as
Hessgf = f
(
Rcmϕ −
Rmϕ
n+m− 1
g
)
. (3.3)
By analyzing the equations satisfied by a weighted vacuum static space, we obtain some
results. First, we have the following.
Proposition 3.3. Let
(
M, g, e−ϕdVg,m
)
be a connected closed weighted vacuum static space.
Then the weighted scalar curvature Rmϕ must be constant.
6 P.T. Ho and J. Shin
Proof. Note that if f is a non-zero constant, then there is nothing to prove. So we assume
that
(
M, g, e−ϕdVg,m
)
is nontrivial. By direct calculation and using Lemma 2.5, we get the
following:
divϕ(∆ϕfg)i = ∇i∆ϕf − (∆ϕf)∇iϕ,
divϕ(Hessgf)i = ∇i∆ϕf +
(
Rcmϕ
)
il
∇lf +
1
m
⟨∇ϕ,∇f⟩∇iϕ,
divϕ
(
fRcmϕ
)
i
=
(
Rcmϕ
)
il
∇lf +
1
2
f∇iR
m
ϕ − 1
m
f(∆ϕϕ)∇iϕ.
Thus, taking the weighted divergence of the first equation in (3.1) yields
divϕ
(
−(∆ϕf)g +Hessgf − fRcmϕ
)
i
=
[
∆ϕf +
1
m
(
⟨∇ϕ,∇f⟩+ f∆ϕϕ
)]
∇iϕ− 1
2
f∇iR
m
ϕ .
Then, by the second equation in (3.1), it reduces to
fdRmϕ = 0.
If f is never zero, Rmϕ must be constant. On the other hand, assume there is some x0 ∈ M
with f(x0) = 0. Then we must have df(x0) ̸= 0. To see this, assume df(x0) = 0, let γ(t) be
a geodesic starting at x, and let h(t) = f
(
γ(t)
)
. It follows from (3.3) that h(t) satisfies the
following linear second order differential equation
h′′(t) = (Hessgf)γ(t) ·
(
γ′(t), γ′(t)
)
=
{(
Rcmϕ −
Rmϕ
n+m− 1
g
)
·
(
γ′(t), γ′(t)
)}
h(t)
with h(0) = f(x0) = 0 and h′(0) = df(γ(0)) · γ′(0) = 0. This implies that h(t) ≡ 0. Thus,
f is zero along γ(t), and by the Hopf–Rinow theorem f vanishes in M , which contradicts the
assumption that f is not constant. Thus df cannot vanish on f−1(0), and 0 is a regular value
of f , which implies that f−1(0) is an (n − 1)-dimensional submanifold of M . Hence, dRmϕ = 0
on an open dense set and hence everywhere in M . ■
Now are ready to prove Proposition 1.1.
Proof of Proposition 1.1. It is well known that if a closed smooth metric measure space satis-
fies Rcmϕ = 0, then it is Ricci-flat and ϕ is constant [24]. So we can assume that
(
M, g, e−ϕdVg,m
)
is a nontrivial weighted vacuum static space. Then it follows from Proposition 3.3 that the
weighted scalar curvature Rmϕ is constant, while (3.2) implies that −∆ϕ −
Rm
ϕ
m+n−1 has nontriv-
ial kernel. Thus Rmϕ ≥ 0. Moreover, if Rmϕ = 0, then f ≡ c for some nonzero constant c. But
then (3.1) gives Rcmϕ = 0 which implies that
(
M, g, e−ϕdVg,m
)
is Ricci-flat and ϕ is constant. ■
The following corollary is a direct consequence of Proposition 1.1.
Corollary 3.4. Let
(
M, g, e−ϕdVg,m
)
be a connected closed smooth metric measure space such
that its weighted scalar curvature Rmϕ is zero and (M, g) is not Ricci-flat. Then
(
M, g, e−ϕdVg,m
)
is not a weighted vacuum static space.
By implicit function theorem, we have the following.
Proposition 3.5. Let
(
M, g, e−ϕdVg,m
)
be a connected closed smooth metric measure space.
Assume that either (i) Rmϕ is not equal to λ(n+m−1) where λ ∈ spec(−∆ϕ), or (ii) R
m
ϕ = 0 and
Rcmϕ ̸≡ 0. Then DR(g,ϕ) is surjective, its kernel splits and R maps any neighborhood of (g, ϕ)
onto a neighborhood of Rmϕ .
Deformation of the Weighted Scalar Curvature 7
Proof. Since DR∗
(g,ϕ) has injective symbol, by Berger–Ebin splitting theorem [4], it suffices to
show that DR∗
(g,ϕ) is injective, for then DR(g,ϕ) will be surjective and its kernel will have a closed
complement, namely Im
(
DR∗
(g,ϕ)
)
, the image of DR∗
(g,ϕ). Injectivity of DR∗
(g,ϕ) follows from
Proposition 1.1. The local surjectivity of R then follows by the implicit function theorem. ■
We have the following example of nontrivial weighted vacuum static space.
Example 3.6. Consider
(
Rn, g0, e−ϕdVg0 ,m
)
, where g0 is the flat metric in Rn = {(x1, . . . , xn) |
xi ∈ R}. Suppose that f = f(x1) and ϕ = ϕ(x1). Then (3.1) can be reduced to the following
two ODEs:
ϕ′f ′ − fϕ′′ +
1
m
f
(
ϕ′
)2
= 0, (3.4)
f ′′ − m− 1
m
ϕ′f ′ +
1
m
fϕ′′ − 1
m
f
(
ϕ′
)2
= 0. (3.5)
Then one can easily see that (f, ϕ) =
(
e−
1
m
x1 , x1
)
satisfies (3.4). So
(
Rn, g0, e−x1dVg0 ,m
)
is
a nontrivial weighted vacuum static space with f = e−
1
m
x1 and the weighted scalar curvature
Rmϕ = −m+1
m . In particular, this example shows that the assumption that M is compact is
needed in Proposition 1.1.
The following proposition gives an existence result for prescribing the weighted scalar curva-
ture.
Proposition 3.7. Let
(
M, g, e−ϕdVg,m
)
be a closed smooth metric measure space such that
its weighted scalar curvature Rmϕ is zero and (M, g) is not Ricci-flat. For any f ∈ C∞(M),
there exists a smooth metric measure space
(
M, g, e−ϕdVg,m
)
such that its weighted scalar
curvature Rm
ϕ
is equal to f .
Proof. It follows from Corollary 3.4 that
(
M, g, e−ϕdVg,m
)
is not a weighted vacuum static
space. In particular, DR∗
(g,ϕ) is injective. Applying Proposition 3.5, there exists ϵ > 0 such that if
∥ψ∥ = ∥Rmϕ − ψ∥ < ϵ, (3.6)
then there exists a smooth metric measure space
(
M, g, e−ϕdVg,m
)
such that its weighted scalar
curvature Rm
ϕ
= ψ. In particular, for any f ∈ C∞(M), we can find a constant c > 0 such that
∥f∥
c
< ϵ,
where ϵ is the number appeared in (3.6). It follows from (3.6) that there exists a smooth met-
ric measure space
(
M, g, e−ϕdVg,m
)
such that its weighted scalar curvature Rm
ϕ
= f
c . Hence,
the smooth metric measure space
(
M, c−1g, c−
n+m
2 e−ϕdVg,m
)
has weighted scalar curvature
cRm
ϕ
= f . ■
Suppose (M, gM ) is an n-dimensional compact space form with sectional curvature 1, and
(N, gN ) is an n-dimensional compact space form with sectional curvature −1. Then M × N
equipped with the product metric gM ⊕ gN is not Ricci-flat. Moreover, the smooth metric
measure space (M ×N, gM ⊕ gN ,dVgM⊕gN ,m), i.e., ϕ ≡ 0, has zero weighted scalar curvature.
It follows from Proposition 3.7 that we can find
(
M × N, g, e−ϕdVg,m
)
such that its weighted
scalar curvature Rm
ϕ
is equal to any prescribed smooth function f .
8 P.T. Ho and J. Shin
4 Weighted vacuum static space
In general relativity, a static spacetime is a four-dimensional Lorentzian manifold which possesses
a timelike Killing field and a spacelike hypersurface which is orthogonal to the integral curves
of this Killing field. In this case coordinates can be chosen so that the metric g is a warped
product of the hypersurface (with metric g) and a time interval, where the warping factor f is
independent of time, i.e.,
g = −f2dt2 + g.
The next proposition shows that the weighted vacuum static space is actually related to static
spacetimes in the weighted sense.
Proposition 4.1. Let
(
M, g, e−ϕdVg,m
)
be a smooth metric measure space. Consider
(
R×M,
g = −f2dt2 + g, e−ϕdVg,m
)
as a smooth metric measure space where ϕ is the pullback of ϕ via
the projection R ×M → M . Then
(
M, g, e−ϕdVg,m
)
is a weighted vacuum static space with a
potential function f if and only if the warped product metric g = −f2dt2+g is Einstein whenever
f ̸= 0 in the weighted sense, i.e., Rcm
ϕ
(g) = kg for some constant k.
Proof. We work on a component of the open set where f is nowhere-zero. Then we have the
following well-known formulas for the curvature tensor of the warped product
(
B ×f F, g =
−f2dt2 + g
)
: For vectors X,Y tangent to the base B and V , W tangent to the 1-dimensional
fiber F , and with Rc = Rc(g),
Rc(X,Y ) = RcB(X,Y )− 1
f
Hessgf(X,Y ),
Rc(X,V ) = 0,
Rc(V,W ) = −∆gf
f
g(V,W ).
Using these, we compute the Bakry–Émery Ricci tensor of g as follows:
Rcm
ϕ
(X,Y ) = Rc(X,Y ) + Hess gϕ(X,Y )− 1
m
dϕ⊗ dϕ(X,Y )
= RcB(X,Y )− 1
f
Hessgf(X,Y ) + Hessgϕ(X,Y )− 1
m
dϕ⊗ dϕ(X,Y )
= Rcmϕ (X,Y )− 1
f
Hessgf(X,Y )
= −
∆ϕf
f
g(X,Y )
=
Rmϕ
n+m− 1
g(X,Y ) =
Rmϕ
n+m− 1
g(X,Y ),
where the fourth equality follows from (3.1) and the fifth equality follows from (3.2). Similarly,
one can easily see that
Rcm
ϕ
(X,V ) = 0,
Rcm
ϕ
(V,W ) = −
∆ϕf
f
g(V,W ) =
Rmϕ
n+m− 1
g(V,W ).
Therefore, the warped product metric g is Einstein in the weighted sense.
The converse can be proved in a similar way. ■
Deformation of the Weighted Scalar Curvature 9
We say that (M, g, f) is a vacuum static space, if (M, g) is a Riemannian manifold and smooth
function f ̸= 0 satisfies the following equation:
Hessgf = f
(
Rc− R
n− 1
g
)
. (4.1)
Proposition 4.2. Let m > 0. Let MR be the space of all closed, connected vacuum static
spaces and Mw be the space of all closed, connected weighted vacuum static spaces. Suppose(
M, g, e−ϕdVg, f
)
∈ MR∩Mw (with the same f), then
(
M, g, e−ϕdVg,m
)
is isometric to a Ricci-
flat manifold with ϕ being constant.
Proof. Suppose f−1(0) ̸= ∅. It is well known that in a vacuum static space, df ̸= 0 on f−1(0)
(cf. [15]). Then f−1(0) is a regular hypersurface in M and hence f ̸= 0 on a dense subset of M .
Combining (3.3) and (4.1), we have
− R
n− 1
f = ∆gf = f
(
m− 1
m+ n− 1
Rmϕ −∆ϕϕ
)
.
Since f is nonzero on a dense set, this implies that
∆ϕϕ =
m− 1
m+ n− 1
Rmϕ +
R
n− 1
. (4.2)
It is well known that if (M, g, f) is vacuum static space, then (M, g) has nonnegative constant
scalar curvature. And by Proposition 3.3, the weighted scalar curvature Rmϕ is also nonnegative
constant. If m ≥ 1, then (4.2) implies ∆ϕϕ ≥ 0. Therefore, R = 0 and ϕ is constant. Thus
Rmϕ = 0. We conclude from Proposition 1.1 that
(
M, g, e−ϕdVg,m
)
is Ricci-flat. Suppose
0 < m < 1. By (1.1), we can rewrite (4.2) as
∆ϕϕ =
mn
(n− 1)(m+ n− 1)
R+
2(m− 1)
m+ n− 1
∆ϕ− (m− 1)(m+ 1)
m(m+ n− 1)
|∇ϕ|2. (4.3)
Integrating (4.3) over dVg, we have∫
M
∆ϕϕdVg =
mn
(n− 1)(m+ n− 1)
∫
M
RdVg −
(m− 1)(m+ 1)
m(m+ n− 1)
∫
M
|∇ϕ|2dVg ≥ 0.
But based on the formula for ∆ϕϕ, this implies that −
∫
M |∇ϕ|2dVg ≥ 0. Therefore, ϕ is constant
and then R = 0. Thus Rmϕ = 0. Again by Proposition 1.1, we conclude that
(
M, g, e−ϕdVg,m
)
is Ricci-flat. ■
Remark 4.3. It would be interesting to study the relation between MR and Mw. One can ask
the following question: If both MR and Mw are nonempty, does that mean MR ∩Mw is also
nonempty? This question is left to future research, and we hope to come back to this later.
We say that
(
M, g, e−ϕdVg,m
)
is locally conformally flat in the weighted sense if for each
p ∈ M , there is a neighborhood U of p on which there is a conformal factor u for which(
e2ug, e(m+n)ue−ϕdVg
)
= (gflat, dVgflat). Just as the local conformal flatness in the Riemannian
manifold
(
Mn, g
)
is equivalent to the vanishing of the Weyl tensor when n ≥ 4, the local con-
formal flatness in the weighted sense has the following characterization: Consider the following
modifications of the curvature tensors:
Pmϕ := Rcϕ − 1
2(m+ n− 1)
Rmϕ g,
Amϕ := Rm− 1
m+ n− 2
Pmϕ ⃝∧ g.
10 P.T. Ho and J. Shin
Here ⃝∧ denoted the Kulkarni–Nomizu product. We call Pmϕ the weighted Schouten tensor,
and Amϕ the weighted Weyl tensor. Then a smooth metric measure space
(
M, g, e−ϕdVg,m
)
,
where n ≥ 3 and m + n ̸= 3, is locally conformally flat in the weighted sense if and only if
Amϕ = 0 (cf. [6, Lemma 6.6]).
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2. We follow the argument in [10] (see also [16]). Taking the covariant
derivative of (3.3) yields
∇i∇j∇kf = Rcϕjk∇if + f∇iRc
ϕ
jk −
Rmϕ
n+m− 1
gjk∇if,
where Rcϕij denotes Rc
m
ϕ (ei, ej). This implies that
Rijkl∇lf = ∇i∇j∇kf −∇j∇i∇kf (4.4)
= Rcϕjk∇if − Rcϕik∇jf + f∇iRc
ϕ
jk − f∇jRc
ϕ
ik −
Rmϕ
n+m− 1
(gjk∇if − gik∇jf).
Note that if
(
M, g, e−ϕdVg,m
)
is locally conformally flat in the weighted sense, then the
weighted Schouten tensor
Pmϕ := Rcmϕ −
Rmϕ
2(m+ n− 1)
g
is a Codazzi tensor, i.e., ∇iP
ϕ
jk = ∇jP
ϕ
ik (see [7, Lemma 3.2]). Since Rmϕ is constant by Proposi-
tion 1.1, this implies that the Bakry–Émery Ricci tensor is a Codazzi tensor. So (4.4) reduces to
Rijkl∇lf = Rcϕjk∇if − Rcϕik∇jf −
Rmϕ
n+m− 1
(gjk∇if − gik∇jf). (4.5)
On the other hand, it follows from Amϕ ≡ 0 that
Rijkl∇lf = − 1
m+ n− 2
(
Rcϕilgjk∇
lf +Rcϕjk∇if − Rcϕik∇jf − Rcϕjlgik∇
lf
)
+
Rmϕ
(n+m− 1)(n+m− 2)
(gjk∇if − gik∇jf). (4.6)
Combining (4.5) and (4.6), we have
−Rcϕilgjk∇
lf +Rcϕjlgik∇
lf
= (n+m− 1)
(
Rcϕjk∇if − Rcϕik∇jf
)
−Rmϕ (gjk∇if − gik∇jf). (4.7)
This shows that ∇f is an eigenvector of Rcϕ, i.e., Rcϕ(X,∇f) = 0 for X ⊥ ∇f .
For any regular value c0 of the function f , consider the level surface Σc0 = f−1(c0). Suppose I
is an open interval containing c0 such that f has no critical points in the open neighborhood
UI = f−1(I) of Σc0 . Then we can express the metric g on UI as
ds2 =
1
|∇f |2
df2 + gf ,
where gf = gab(f, θ)dθ
adθb is the induced metric and θ =
(
θ1, . . . , θn
)
is any local coordinates
system on Σc0 .
Deformation of the Weighted Scalar Curvature 11
On the other hand, for any vector field X ⊥ ∇f , we have
∇X
(
|∇f |2
)
= 2∇2f(∇f,X) = 2f
(
Rcϕ(∇f,X)−
Rmϕ
n+m− 1
g(∇f,X)
)
= 0,
where the second equality follows from (3.3) and the last equality follows from the fact that∇f is
an eigenvector of Rcϕ. Hence, |∇f |2 is constant on any regular value surface Σc = f−1(c) ⊂ UI ,
which are all diffeomorphic to Σc0 . This allows us to make the change of variable by setting
r(x) =
∫
df
|∇f |
,
so that we can further express the metric g on UI as
ds2 = dr2 + gab(r, θ)dθ
adθb.
Let ∇r = ∂
∂r . Then |∇r| = 1 and ∇f = f ′(r) ∂∂r on UI . Note that f ′(r) does not change sign
on UI because f has no critical points there. Thus, we may assume I = (α, β) with f ′(r) > 0
for r ∈ (α, β). It is also easy to check that
∇ ∂
∂r
∂
∂r
= 0, (4.8)
so integral curves to ∇r are geodesics.
Next, it follows from (3.3) and (4.8) that
f
(
Rcϕrr −
Rmϕ
n+m− 1
)
= ∇2f
(
∂
∂r
,
∂
∂r
)
= f ′′(r). (4.9)
We can therefore conclude that Rcϕrr is also constant on Σc ⊂ UI . Moreover, the second funda-
mental form of Σc is given by
hab =
∇a∇bf
|∇f |
=
f
f ′
(
Rcϕab −
Rmϕ
n+m− 1
gab
)
. (4.10)
On the other hand, by (4.7), we have
Rcϕab =
1
n+m− 1
(
Rmϕ − Rcϕrr
)
gab. (4.11)
Combining (4.10) and (4.11), we have
hab = − 1
n+m− 1
f
f ′
Rcϕrrgab. (4.12)
In particular, Σc is umbilical and its mean curvature is given by
H = − n− 1
n+m− 1
f
f ′
Rcϕrrgab,
which is again constant along Σc.
Now, we fix a local coordinates system(
x1, x2, . . . , xn
)
=
(
r, θ2, . . . , θn
)
in UI , where
(
θ2, . . . , θn
)
is any local coordinates system on the level surface Σc0 , and indices
a, b, c, . . . range from 2 to n. Then, computing in this local coordinates system we obtain that
hab = −⟨∂r,∇a∂b⟩ = −
〈
∂r,Γ
1
ab∂r
〉
= −Γ1
ab.
12 P.T. Ho and J. Shin
But the Christoffel symbol Γ1
ab is given by
Γ1
ab =
1
2
g11
(
−∂gab
∂r
)
= −1
2
∂gab
∂r
.
Hence, we get
∂gab
∂r
= − 2
n+m− 1
f
f ′
Rcϕrrgab = − 2
n+m− 1
f
f ′
(
f ′′
f
+
Rmϕ
n+m− 1
)
gab,
where the second equality follows from (4.9). Thus we can see that in any open neighborhood
Uβα = f−1((α, β)) of Σc in which f has no critical points, the metric g can be expressed as
ds2 = dr2 + w(r)2ḡ,
where
(
θ2, . . . , θn
)
is any local coordinates system on Σc, ḡ = gab(r0, θ)dθ
adθb is the induced
metric on Σc = r−1(r0), and the warping function w(r) satisfying
w′
w
= − 1
n+m− 1
f ′′
f ′
−
Rmϕ
(n+m− 1)2
f
f ′
.
Furthermore, from the Gauss equation, one can see that the sectional curvatures of (Σc, ḡ) are
given by
RΣ
abab = Rabab + haahbb − h2ab
=
1
n+m− 2
(
Rcϕaa +Rcϕbb
)
−
Rmϕ
(n+m− 1)(n+m− 2)
+
1
(n+m− 1)2
(
f
f ′
Rcϕrr
)2
=
Rmϕ − 2Rcϕrr
(n+m− 1)(n+m− 2)
+
1
(n+m− 1)2
(
f
f ′
Rcϕrr
)2
for a, b = 2, . . . , n, where the second equality follows from Amϕ ≡ 0 and (4.12), and the third
equality follows from (4.11). Since all the terms on the right-hand side are constant on Σc, we
conclude that the sectional curvatures of (Σc, ḡ) are constant. ■
5 Rigidity phenomena of flat manifolds
and prescribing the weighted scalar curvature
First we prove Theorem 1.3.
Proof of Theorem 1.3. Let (F, gF ) be a m-dimensional closed flat Riemannian manifold.
Since g is flat and ϕ is constant,
(
M × F, g1 := g + e−2ϕ/mgF
)
is a (n +m)-dimensional closed
flat Riemannian manifold. By [15, Theorem B], there is a ϵ1 such that if a Riemannian met-
ric h on M × F has nonnegative scalar curvature and ∥g1 − h∥C2(M×F,g1) < ϵ1, then h is flat.
Let (g, ϕ) be a smooth metric measure structure on M with Rmϕ ≥ 0. Then g2 := g + e−
2ϕ
m gF is
a Riemannian metric on M × F such that its scalar curvature Rg2 = Rmϕ is nonnegative. And
one can easily see that
∥g1 − g2∥C2(M×F,g1) ≤ C1
∥∥(g, ϕ)− (g, ϕ)
∥∥
C2(M,g)
for some constant C1 = C1
(
M, g, ϕ
)
. Thus if
∥∥(g, ϕ) − (g, ϕ)
∥∥
C2(M,g)
< ϵ1
C1
then we have
∥g1 − g2∥C2(M×F,g1) < ϵ1 which implies that g2 = g + e−
2ϕ
m gF must be flat. Since gF is flat, this
implies that g is flat and ϕ is constant. ■
Deformation of the Weighted Scalar Curvature 13
In the remainder of this section, we consider the problem of prescribing the weighted scalar
curvature on smooth metric measure spaces. More precisely, given a smooth function f in M ,
we want to find a smooth metric measure space
(
M, g, e−ϕdVg,m
)
such that the weighted scalar
curvature is equal to f . In particular, we are going to prove Theorem 1.4. To this end, we first
recall the following approximation lemma.
Lemma 5.1 ([20, Theorem 2.1]). Let f1, f2 ∈ C∞(M). If min f1 ≤ f2 ≤ max f1 in M , then
given any positive ϵ, there is a diffeomorphism φ of M such that, for p > n, we have that
∥f1 ◦ φ− f2∥Lp(M) < ϵ.
We also need the following.
Proposition 5.2. Let f ∈ Lp(M) with p > n. Suppose that DR∗
(g0,ϕ0)
is injective. There is
an η > 0 such that if
∥f −Rmϕ0∥Lp(M) < η,
then there is a metric measure (g1, ϕ1) ∈ M2,p×W 2,p such that R(g1, ϕ1) = f . Moreover, (g1, ϕ1)
is smooth in any open set where f is smooth.
Proof. We consider the following operator S : U ⊂W 4,p(M) → Lp(M) defined by
S(u) = Rm
ϕ
,
where(
g, ϕ
)
= (g0, ϕ0) +DR∗
(g0,ϕ0)
(u)
and U is a sufficiently small neighborhood of zero inW 4,p. We claim that S′(0) is an isomorphism
restricted to a small neighborhood in W 4,p norm. In fact, S(0) = R(g0, ϕ0) and
S′(0)v = DR(g0,ϕ0)DR∗
(g0,ϕ0)
v.
Hence, kerS′(0) = kerDR(g0,ϕ0)DR∗
(g0,ϕ0)
⊆ kerDR∗
(g0,ϕ0)
= {0}, which implies kerS′(0) = {0}.
It follows from the implicit function theorem that S maps a neighborhood of zero in W 4,p onto
a neighborhood S(0) = R(g0, ϕ0) in L
p(M). Thus there is an η > 0 such that if
∥f1 −Rmϕ0∥Lp(m) < η,
then there exists a solution (g1, ϕ1) = (g0, ϕ0)+DR∗(u) ofR(g1, ϕ1) = f . Using elliptic regularity
and a bootstrap argument, we have that if f is smooth, then u is smooth. ■
Now we are ready to prove Theorem 1.4.
Proof of Theorem 1.4. We perturb (g0, ϕ0) slightly, if necessary, to obtain (g1, ϕ1) such that
the weighted scalar curvature R(g1, ϕ1) is not constant with c1minK < R(g1, ϕ1) < c1maxK
in M , where c1 > 0 is constant. It follows from Proposition 1.1 that
(
M, g1, e
−ϕ1dVg1 ,m
)
is not
a weighted vacuum static space. In particular, DR∗
(g1,ϕ1)
is injective. It follows from Lemma 5.1
that there is a diffeomorphism φ of M such that ∥c1K ◦ φ − k1∥p < η, where p > dimM .
Since DR∗
(g1,ϕ1)
is injective, we can apply Proposition 5.2 to conclude that there is a smooth
metric measure (g2, ϕ2) such that R(g2, ϕ2) = c1K ◦ φ. If we let (g, ϕ) =
(
φ−1
)∗
(c1(g2, ϕ2)),
the smooth metric measure space
(
M, g, e−ϕdVg,m
)
has weighted scalar curvature being equal
to K. ■
14 P.T. Ho and J. Shin
Acknowledgements
The authors would like to thank the referees for comments and suggestions, which improve the
presentation of this paper. The first author was supported by the National Science and Technol-
ogy Council (NSTC), Taiwan, with grant Number: 112-2115-M-032-006-MY2, and the second
author was supported by a KIAS Individual Grant (SP070701) via the Center for Mathematical
Challenges at Korea Institute for Advanced Study.
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1 Introduction
2 Smooth metric measure space
3 Deformation of the weighted Scalar curvature
4 Weighted vacuum static space
5 Rigidity phenomena of flat manifolds and prescribing the weighted scalar curvature
References
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| id | nasplib_isofts_kiev_ua-123456789-212044 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T15:33:05Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ho, Pak Tung Shin, Jinwoo 2026-01-23T10:11:32Z 2023 Deformation of the Weighted Scalar Curvature. Pak Tung Ho and Jinwoo Shin. SIGMA 19 (2023), 087, 15 pages 1815-0659 2020 Mathematics Subject Classification: 53C21; 53C23 arXiv:2311.02359 https://nasplib.isofts.kiev.ua/handle/123456789/212044 https://doi.org/10.3842/SIGMA.2023.087 Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal ²ϕ-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particular, we define a weighted vacuum static space and study locally conformally flat weighted vacuum static spaces. We then prove some stability results of the weighted scalar curvature on flat spaces. Finally, we consider the prescribed weighted scalar curvature problem on closed smooth metric measure spaces. The authors would like to thank the referees for their comments and suggestions, which have improved the presentation of this paper. The first author was supported by the National Science and Technology Council (NSTC), Taiwan, with grant Number 112-2115-M-032-006-MY2, and the second author was supported by a KIAS Individual Grant (SP070701) via the Center for Mathematical Challenges at Korea Institute for Advanced Study. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformation of the Weighted Scalar Curvature Article published earlier |
| spellingShingle | Deformation of the Weighted Scalar Curvature Ho, Pak Tung Shin, Jinwoo |
| title | Deformation of the Weighted Scalar Curvature |
| title_full | Deformation of the Weighted Scalar Curvature |
| title_fullStr | Deformation of the Weighted Scalar Curvature |
| title_full_unstemmed | Deformation of the Weighted Scalar Curvature |
| title_short | Deformation of the Weighted Scalar Curvature |
| title_sort | deformation of the weighted scalar curvature |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212044 |
| work_keys_str_mv | AT hopaktung deformationoftheweightedscalarcurvature AT shinjinwoo deformationoftheweightedscalarcurvature |