## Curvature-Dimension Condition Meets Gromov’s n-Volumic Scalar Curvature

Jialong DENG

Mathematisches Institut, Georg-August-Universit¨at, G¨ottingen, Germany E-mail: jialong.deng@mathematik.uni-goettingen.de

Received July 29, 2020, in final form January 23, 2021; Published online February 05, 2021 https://doi.org/10.3842/SIGMA.2021.013

Abstract. We study the properties of then-volumic scalar curvature in this note. Lott–

Sturm–Villani’s curvature-dimension condition CD(κ, n) was showed to imply Gromov’s n-volumic scalar curvature≥nκunder an additionaln-dimensional condition and we show the stability ofn-volumic scalar curvature≥κwith respect to smGH-convergence. Then we propose a new weighted scalar curvature on the weighted Riemannian manifold and show its properties.

Key words: curvature-dimension condition; n-volumic scalar curvature; stability; weighted scalar curvature Scα,β

2020 Mathematics Subject Classification: 53C23

### 1 Introduction

The concept of lower bounded curvature on the metric space or the metric measure space has
evolved to a rich theory due to Alexandrov’s insight. The stability of Riemannian manifolds
with curvature bounded below is another deriving force to extend the definition of the curvature
bounded below to a broader space. However, the scalar curvature (of Riemannian metrics)
bounded below was yet absent from this picture. Gromov proposed a synthetic treatment of
scalar curvature bounded below, which was called then-volumic scalar curvature bounded below,
and offered some pertinent conjectures in [18, Section 26]. Motivated by the CD(κ, n) condition,
we add ann-dimension condition to the Gromov’s definition and introduce the definition of Sc_{α,β}
on the smooth metric measure space. Details will be given later.

Theorem 1.1. Assume that the metric measure space (X^{n}, d, µ) satisfies n-dimensional con-
dition and the curvature-dimension condition CD(κ, n) for κ ≥ 0 and n ≥ 2, then (X^{n}, d, µ)
satisfies Sc^{vol}^{n}(X^{n})≥nκ.

Theorem 1.2. If compact metric measure spaces (X_{i}^{n}, di, µi) with Sc^{vol}^{n}(X_{i}^{n}) ≥ κ ≥ 0 and
SC-radius r_{x}^{n}

i ≥ R > 0 and (X_{i}^{n}, d_{i}, µ_{i}) strongly measured Gromov–Hausdorff converge to the
compact metric measure space (X^{n}, d, µ) with n-dimensional condition, then X^{n} also satisfies
Sc^{vol}^{n}(X^{n})≥κ and the SC-radius rX^{n} ≥R.

Theorem 1.3. Let M^{n}, g,e^{−f}dVolg

be the closed smooth metric measure space withScα,β >0, then we have the following conclusions:

1. If M^{n} is a spin manifold, α∈Rand β ≥ ^{|α|}_{4}^{2}, then the harmonic spinors of M^{n} vanish.

2. If the dimension n≥3, α ∈Rand β ≥ ^{(n−2)|α|}_{4(n−1)}^{2}, then there is a metricg˜ conformal to g
with positive scalar curvature.

This paper is a contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday. The full collection is available athttps://www.emis.de/journals/SIGMA/Gromov.html

3. If the dimension n ≥ 3, α = 2, β ≥ ^{n−2}_{n−1} and N^{n−1},¯g

is the compact L_{f}-stable min-
imal hypersurface of M^{n}, g,e^{−f}dVol_{g}

, then there exists a PSC-metric conformal to g¯
on N^{n−1}, where g¯ is the induced metric ofg on N^{n−1}.

4. Assume M^{n} is a spin manifold and there exists a smooth1-contracting map h: (M^{n}, g)→
(S^{n}, g_{st}) of non-zero degree. If α ∈ R , β ≥ ^{|α|}_{4}^{2} and Sc_{α,β} ≥ n(n−1), then h is an
isometry between the metrics g and g_{st}.

The paper is organized as follows. In Section 2, we introduce the notions and show that
CD(κ, n) implies Sc^{vol}^{n} ≥(n−1)κ. In Section3, we show the stability of spaces with Sc^{vol}^{n} ≥κ.

In Section 4, we present the properties of the smooth metric measure space with Scα,β >0.

### 2 CD meets n-volumic scalar curvature

The n-dimensional Aleksandrov space with curvature ≥ κ equipped with the volume-measure satisfies Lott–Villani–Sturm’s weak curvature-dimension condition for dimension n and curva- ture (n−1)κ, i.e., CD((n−1)κ, n), was shown by Petrunin forκ= 0 (and said that for general curvature ≥ κ the result followed in a similar way) [32] and then Zhang–Zhu investigated the general case [43]. We will modify Gromov’s definition of n-volumic scalar curvature bounded below in [18, Section 26] to fill the picture, which means Lott–Sturm–Villani’s Ricci curvature

≥0 implies Gromov’s scalar curvature≥0.

The metric measure space (mm-space)X = (X, d, µ) means that dis the complete separable
length metric on X and µ is the locally finite full support Borel measure onX equipped with
its Borel σ-algebra. Say that an mm-spaceX= (X, d, µ) is locally volume-wise smaller (or not
greater) than another such spaceX^{0} = (X^{0}, d^{0}, µ^{0}) and write X <_{vol}X^{0} (X ≤_{vol} X^{0}), if all-balls
in X are smaller (or not greater) than the -balls in X^{0}, µ(B(x)) < µ^{0}(B(x^{0}))(µ(B(x)) ≤
µ^{0}(B_{}(x^{0})), for allx∈X, x^{0} ∈X^{0} and the uniformly smallwhich depends on X andX^{0}.

From now on, the Riemannian 2-sphere S^{2}(γ), d_{S},vol_{S}

is endowed with round metric such
that the scalar curvature equal to 2γ^{−2}, R^{n−2}, dE,volE

is endowed with Euclidean metric with
flat scalar curvature and the product manifold S^{2}(γ)×R^{n−2} is endowed with the Pythagorean
product metrics dS×E :=

q

d^{2}_{S}+d^{2}_{E} and the volume volS×E := volS⊗volE.

Thus, we have S^{2}(γ) <_{vol} R^{2}. If 0 < γ1 < γ2, then S^{2}(γ1) <_{vol} S^{2}(γ2). Furthermore,
S^{2}(γ)×R^{n−2} <_{vol}R^{n}. If 0< γ_{1} < γ_{2}, thenS^{2}(γ_{1})×R^{n−2} <_{vol} S^{2}(γ_{2})×R^{n−2}.

Definition 2.1 (Gromov’s n-volumic scalar curvature). Gromov’s n-volumic scalar curvature
of X is bounded below by 0 forX = (X, d, µ) if X is locally volume-wise not greater thanR^{n}.
Gromov’s n-volumic scalar curvature of X bounds from below by κ > 0 for X = (X, d, µ)
ifXis locally volume-wise smaller thanS^{2}(γ)×R^{n−2} for allγ >

q2

κ, i.e.,X <_{vol}S^{2}(γ)×R^{n−2}
and γ >

q2

κ, whereS^{2}(γ)×R^{n−2}= S^{2}(γ)×R^{n−2}, dS×E,volS×E

.

Then-volumic scalar curvature is sensitive to the scaling of the measure, but the curvature condition CD(κ, n) of Lott–Villani–Sturm [38, Definition 1.3] is invariant up to scalars of the measure only [38, Proposition 1.4(ii)]. Therefore, the n-dimensional condition needs to be put into the definition of Gromov’sn-volumic scalar curvature. In fact, then-dimensional condition is the special case of Young’s point-wise dimension in dynamical systems [41, Theorem 4.4].

Definition 2.2 (n-dimensional condition). For given positive natural numbern, the mm-space X = (X, d, µ) satisfies then-dimensional condition if

r→0lim

µ(B_{r}(x))
volE(Br(R^{n})) = 1

for every x∈X, whereBr(R^{n}) is the closed r-ball in the Euclidean spaceR^{n} and theBr(x) is
the closed r-ball with the centerx∈X.

From now on, the superscript ofnin the space X^{n} means the mm-space (X^{n}, d, µ) satisfies
n-dimensional condition.

Note that a closed smoothn-manifoldM^{n}(n≥3) admits a Riemannian metric with constant
negative scalar curvature and a Riemannian metric of non-negative scalar curvature which is
not identically zero, then by a conformal change of the metric we get a metric of positive
scalar curvature according to Kazdan–Warner theorem [25]. Furthermore, if there is a scalar-
flat Riemannian metric g on M^{n}, but g is not Ricci-flat metric, then g can be deformed to
a metric with positive scalar curvature according to Kazdan theorem [24, Theorem B] or by
using Ricci-flow with an easy argument. Hence we will focus more on promoting the positive
scalar curvature to positive n-volumic scalar curvature.

Definition 2.3(n-volumic scalar curvature). AssumeX^{n}= (X^{n}, d, µ) is the compact mm-space
and satisfies the n-dimensional condition, we call

1. then-volumic scalar curvature ofX^{n}is positive, i.e., Sc^{vol}^{n}(X^{n})>0, if there existsr_{X}^{n} >0
such that the measures of-balls inX^{n} are smaller than the volumes of -balls inR^{n} for
0< ≤rX^{n}.

2. then-volumic scalar curvature ofX^{n} is bounded below by 0, i.e., Sc^{vol}^{n}(X^{n})≥0, if there
exists r_{X}^{n} >0 such that the measures of -balls inX^{n} are not greater than the volumes
of -balls in R^{n} for 0< ≤rX^{n}.

The rX^{n} is called scalar curvature radius (SC-radius) of X^{n} for Sc^{vol}^{n}(X^{n})≥0.

3. then-volumic scalar curvature ofX^{n}is bounded below byκ >0, i.e., Sc^{vol}^{n}(X^{n})≥κ >0,
if, for any γ withγ >

q2

κ, there existsr_{X}^{n}_{,γ} >0 such that the measures of-balls in X^{n}
are smaller than the volumes of -balls in S^{2}(γ)×R^{n−2} for 0< ≤rX^{n},γ.

We callr_{X}^{n} := inf

γ>

q2 κ

r_{X}^{n}_{,γ} is the SC-radius of X^{n} for Sc^{vol}^{n}(X^{n})≥κ >0.

In particular, we will focus on the case of inf

γ>

q2 κ

r_{X}^{n}_{,γ} 6= 0 for stability in Section3.

If the mm-spaceX^{n} is locally compact, then the definition of then-volumic scalar curvature
bounded below only modifies the definition of the r_{X}^{n}_{,γ} > 0 to a positive continuous function
of X^{n}.

Two mm-spaces (X^{n}, d, µ) and (X_{1}^{n}, d1, µ1) are isometric if there exists a one-to-one map
f:X^{n}→X_{1}^{n} such thatd_{1}(f(a), f(b)) =d(a, b) foraandbare inX^{n}andf∗µ=µ_{1}, wheref∗µis
the push-forward measure, i.e., f∗µ(U) =µ f^{−1}(U)

for a measureable subset U ⊂X_{1}^{n}. IfX^{n}
satisfies Sc^{vol}^{n}(X^{n}) ≥ κ ≥ 0, then each mm-space (X_{1}^{n}, d_{1}, µ_{1}) that is isometric to (X^{n}, d, µ)
also satisfies Sc^{vol}^{n}(X_{1}^{n})≥κ≥0.

Proposition 2.4. Letgbe aC^{2}-smooth Riemannian metric on a closed orientedn-manifoldM^{n}
with induced metric measure space (M^{n}, d_{g},dVol_{g}), then the scalar curvature of g is positive,
Scg>0, if and only if Sc^{vol}^{n}(M^{n})>0, andScg≥κ >0 if and only if Sc^{vol}^{n}(M^{n})≥κ >0.

Proof . For aC^{2}-smooth Riemannian metricg, one has
dVolg(Br(x)) = volE(Br(R^{n}))

1− Scg(x)

6(n+ 2)r^{2}+O r^{4}

forBr(x)⊂M^{n} asr→0. Hence (M^{n}, dg,dVolg) satisfies then-dimensional condition.

If we have Scg >0, then, sinceM^{n} is compact, there existsrM^{n} >0, so that dVolg(Br(x))<

vol_{E}(B_{r}(R^{n})) for all 0 < r ≤ r_{M}^{n}. On the other hand, if there exists r_{M}^{n} > 0 such that
dVol_{g}(B_{r}(x))<vol_{E}(B_{r}(R^{n})) for all 0< r≤r_{M}^{n}, then Sc_{g} must be greater than 0.

If Sc^{vol}^{n}(M^{n}) ≥ κ > 0, then Scg ≥ κ > 0. Otherwise, assume there exist small > 0 such
that Sc_{g}≥κ− >0. That means that there exists a pointx_{0} inM^{n}such that Sc_{g}(x_{0}) =κ−,
asM^{n} is compact and the scalar curvature is a continuous function onM^{n}. Thus, we can find
a smallr-ballBr(x0) such that the volume ofBr(x0) is greater than the volume of ther-ball in
theS^{2}(γ)×R^{n−2} forγ =q

2
κ−^{}

2

, which is a contradiction.

On the other hand, Scg ≥κ >0 implies Sc^{vol}^{n}(M^{n})≥κ >0. Assume Scg(x1) =κ for some
x_{1}∈M^{n}, then there existsr_{1} such that dVol_{g}(B_{r}_{1}(x))≤dVol_{g}(B_{r}_{1}(x_{1})) for r_{1}-balls inM^{n}and

dVolg(Br(x1)) = volE(Br(R^{n}))

1− κ

6(n+ 2)r^{2}+O r^{4}

as r → 0. Thus, for any γ with γ >

q2

κ, there exists r_{M}^{n}_{,γ} > 0 such that the measures of
-balls in M^{n} are smaller than the volumes of -balls in S^{2}(γ)×R^{n−2} for 0< ≤r_{M}^{n}_{,γ}, i.e.,

Sc^{vol}^{n}(M^{n})≥κ >0.

Therefore, we haveS^{n}κ

n−1 <_{vol} S^{2}(γ)×R^{n−2} for all γ >

q 2

nκ. HereS^{n}κ

n−1 is the Riemannian
manifold S^{n} with constant sectional curvature _{n−1}^{κ} .

Remark 2.5. For a closed smooth Riemannian manifold (M^{n}, g), Sc^{vol}^{n}(M^{n}) ≥ 0 implies
Scg≥0. Otherwise, there exists a point in M^{n}such that the scalar curvature is negative, then
the volume of small ball will be greater than the volume of the small ball in Euclidean space,
which is a contradiction.

On the other hand, one can consider the case of the scalar-flat metric, i.e., Scg ≡ 0. If g
is a strongly scalar-flat metric, meaning a metric with scalar curvature zero such that M^{n} has
no metric with positive scalar curvature, then g is also Ricci flat according to Kazdan theorem
above. Thus, we have

dVol_{g}(B_{r}(x)) = vol_{E}(B_{r}(R^{n}))

"

1− kRie(x)k^{2}_{g}

120(n+ 2)(n+ 4)r^{4}+O r^{6}

#

forBr(x)⊂M^{n}asr→0 [14, Theorem 3.3]. Here Rie is the Riemannian tensor. Therefore, ifg
is a not flat metric, then M^{n} <_{vol} R^{n}. If g is a flat metric, then M^{n} ≤_{vol} R^{n}. Thus Sc_{g} ≥0
implies Sc^{vol}^{n}(M^{n})≥0 for a strongly scalar-flat metric g.

However, Scg ≥0 may not imply Sc^{vol}^{n}(M^{n}) ≥0. There are a lot of scalar-flat metrics but
not strongly scalar flat metrics, i.e., Sc_{g} ≡ 0 but not Ricc_{g} 6= 0. For instance, the product
metric on S^{2}(1)×Σ, where Σ is a closed hyperbolic surface, is the scalar-flat metric, but not
the Ricci-flat metric. For those metrics, we have

dVolg(Br(x)) = volE(Br(R^{n}))

"

1 +−3kRie(x)k^{2}_{g}+ 8kRicc(x)k^{2}_{g}

360(n+ 2)(n+ 4) r^{4}+O r^{6}

#

forBr(x)⊂M^{n} asr→0 [14, Theorem 3.3]. If 8kRicc(x)k^{2}_{g} >−3kRie(x)k^{2}_{g}for some point, then
Sc_{g}≥0 does not imply Sc^{vol}^{n}(M^{n})≥0.

Theorem 2.6. Assume that the mm-space (X^{n}, d, µ) satisfies n-dimensional condition and
the curvature-dimension condition CD(κ, n) for κ ≥ 0 and n ≥ 2, then (X^{n}, d, µ) satisfies
Sc^{vol}^{n}(X^{n})≥nκ.

Proof . In fact, one only needs the generalized Bishop–Gromov volume growth inequality, which
is implied by the curvature-dimension of X^{n} [38, Theorem 2.3].

(i) Ifκ= 0, then
µ(B_{r}(x))
µ(BR(x)) ≥r

R n

for all 0< r < R. That is µ(Br(x))

vol_{E}(B_{r}(R^{n})) = µ(Br(x))

α(n)r^{n} ≥ µ(BR(x))

α(n)R^{n} = µ(BR(x))
vol_{E}(B_{R}(R^{n})),
where α(n) = ^{vol}^{E}^{(B}_{r}n^{r}^{(R}^{n}^{))}. Combining then-dimensional condition,

r→0lim

µ(B_{r}(x))

vol_{E}(B_{r}(R^{n})) = 1,
that implies Sc^{vol}^{n}(X)≥0.

(ii) Ifκ >0, then
µ(Br(x))
µ(B_{R}(x)) ≥

R_{r}

0

sin tq

κ (n−1)

n−1

dt RR

0

sin tq _{κ}

(n−1)

n−1

dt for all 0< r≤R≤π

q(n−1) κ .

Since the scalar curvature of the product manifoldS^{2}(γ)×R^{n−2}isnκ, whereγ =
q 2

nκ, then
there existsC_{1}, C_{2} >0 such that

1− nκ

6(n+ 2)r_{1}^{2}−C_{2}r^{4}_{1} ≤volS×E^(B_{r}_{1}(y)) := volS×E(B_{r}_{1}(y))

vol_{E}(B_{r}_{1}(R^{n})) ≤1− nκ

6(n+ 2)r_{1}^{2}+C_{2}r_{1}^{4},
fory ∈S^{2}(γ)×R^{n−2} and r1 ≤C1, where C1,C2 are decided by the product manifold S^{2}(γ)×
R^{n−2}.

Let

µ(B^_{r}(x)) := µ(Br(x))
vol_{E}(B_{r}(R^{n}))
and

f(r) :=

Rr 0

sin tq _{κ}

(n−1)

n−1

dt
vol_{E}(B_{r}(R^{n})) ,

then the generalized Bishop–Gromov inequality can be re-formulated as
µ(B^_{R}(x))≤µ(B^_{r}(x))f(R)

f(r) for all 0< r < R≤π

q(n−1)

κ . The asymptotic expansion off(r) is f(r) =

1
nr^{n} _{κ}

(n−1)

^{(n−1)}_{2}

− _{6(n+2)}^{(n−1)}r^{n+2} _{κ}

(n−1)

^{n+1}_{2}

+O r^{n+4}
volE(Br(R^{n}))

asr →0. Thus, the asymptotic expansion of ^{f}_{f}^{(R)}_{(r)} is
f(R)

f(r) = 1−_{6(n+2)}^{nκ} R^{2}+O R^{4}
1−_{6(n+2)}^{nκ} r^{2}+O r^{4}

asR→0, r→0. Then-dimensional condition, lim

r→0

µ(B^_{r}(x)) = 1, implies that
µ(B^_{R}(x))≤1− nκ

6(n+ 2)R^{2}+O R^{4}

as R → 0. Therefore, for any κ^{0} with 0 < κ^{0} < κ, there exists _{κ}^{0} > 0 such that for any
0< R≤_{κ}^{0}, we have

µ(B^R(x))<volS×E^(BR(y)),
where volS×E^(BR(y)) = ^{vol}_{vol}^{S×E}^{(B}^{R}^{(y))}

E(BR(R^{n})) is defined as before, the ballsBR(y) are in S^{2}(γ)×R^{n−2}
and γ =

q 2

nκ^{0}. That isX^{n}<_{vol} S^{2}(γ)×R^{n−2}, for allγ >

q 2

nκ, i.e., Sc^{vol}^{n}(X^{n})≥nκ.

In fact, one has the classical Bishop inequality by adding then-dimensional condition to the generalized Bishop–Gromov volume growth inequality. It means that

ifκ= 0, µ(B_{R}(x))≤vol_{E}(B_{R}(R^{n})) for allR >0,

ifκ >0, µ(B_{R}(x))≤vol_{S}^{n} B_{R} S^{n}κ
n−1

for 0< R≤π

q(n−1)
κ .
In other words, ifκ= 0,X^{n}≤_{vol}R^{n}. Ifκ >0,X^{n}≤_{vol} S^{n}^{κ}

n−1. We haveS^{n}^{κ}

n−1 <volS^{2}(γ)×R^{n−2}
for all γ >

q 2

nκ. Then X^{n}<_{vol} S^{2}(γ)×R^{n−2} for all γ >

q 2 nκ.

Thus, we also get Sc^{vol}^{n}(X^{n})≥nκ.

Remark 2.7. Hence the mm-space (X^{n}, d, µ) with Sc^{vol}^{n}(X^{n}) ≥ nκ includes the mm-spaces
that satisfies n-dimensional condition and the generalized Bishop–Gromov volume growth in-
equality as stated in the proof, e.g., the mm-spaces with the Riemannian curvature condition
RCD(κ, n) [2] or with the measure concentration property MCP(κ, n) [29].

Question 2.8. Let Al^{n}(1)be an orientable compactn-dimensional Aleksandrov space with cur-
vature≥1, then do all continuous mapsφfromAl^{n}(1) to the sphereS^{n} with standard metric of
non-zero degree satisfy Lip(φ)≥C(n)? Here Lip(φ) is the Lipschitz constant of φ, φmaps the
boundary of Al^{n}(1) to a point inS^{n} andC(n)is a constant depending only on the dimension n.

Question 2.9. Assume the compact mm-space (X, d, µ) satisfies the curvature-dimension con-
dition CD(n−1, n), n-dimensional condition and the covering dimension is also n, then do all
continuous maps φ from(X, d, µ) to the sphere S^{n} with standard metric, where φ is non-trivial
in the homotopy class of maps, satisfy Lip(φ) ≥ C_{1}(n), where C_{1}(n) is a constant depending
only on n?

Remark 2.10. The questions above are inspired by Gromov’s spherical Lipschitz bounded theorem in [19, Section 3] and the results above. The finite covering dimension is equal to the cohomological dimension over integer ringZ for the compact metric space according to the Alexandrov theorem. The best constant of C(n) and C1(n) would be 1 if both questions have positive answers.

Proposition 2.11 (quadratic scaling). Assume the compact mm-space (X^{n}, d, µ) satisfies
Sc^{vol}^{n}(X^{n}) ≥ κ > 0, then Sc^{vol}^{n}(λX^{n}) ≥ λ^{−2}κ > 0 and r_{λX}^{n} = λr_{X}^{n} for all λ > 0, where
λX^{n}:= (X^{n}, λ·d, λ^{n}·µ).

Proof . First, we will show that then-dimensional condition is stable under scaling. Let d^{0} :=

λ·d,µ^{0} :=λ^{n}·µ,B_{r}^{0}(x) be anr-ball in the (X^{n}, d^{0}), andBr(x) be anr-ball in the (X^{n}, d), then
B^{0}_{r}(x) =B^{r}

λ(x) as the subset in the X^{n}. One has

r→0lim

µ^{0}(B_{r}^{0}(x))

vol_{E}(B_{r}(R^{n})) = lim

r→0

µ^{0}(B^{r}

λ(x))

vol_{E}(B_{r}(R^{n})) = lim

r→0

λ^{n}·µ(B^{r}

λ(x))
λ^{n}·vol_{E}(B^{r}

λ(R^{n})) = 1,
then λX^{n} satisfies then-dimensional condition.

Sinceλ· S^{2}(γ)×R^{n−2}

=λ·S^{2}(γ)×λ·R^{n−2} =S^{2}(λγ)×λ·R^{n−2}, we have
λ·X^{n}<vol λ· S^{2}(γ)×R^{n−2}

=S^{2}(λγ)×λ·R^{n−2}
for all λγ >

q 2

λκ and 0 < ≤ λr_{X}^{n}. That means Sc^{vol}^{n}(λX^{n}) ≥ λ^{−2}κ > 0 and rλ·X^{n} =

λrX^{n}.

We also have Sc^{vol}^{n}(λX^{n})≥0 (>0), if Sc^{vol}^{n}(X^{n})≥0 (>0).

Remark 2.12. Since then-dimensional condition and definition of n-volumic scalar curvature is locally defined, we have the following construction.

Global to local: Let the locally compact mm-space (X^{n}, d, µ) satisfy Sc^{vol}^{n}(X^{n}) ≥κ≥0
andY^{n}⊂Xbe an open subset. Then, if (Y^{n}, dY) is a complete length space, (Y^{n}, dY, µxY)
satisfies Sc^{vol}^{n}(Y^{n})≥κ≥0 andr_{Y}^{n} =r_{X}^{n}. Whered_{Y} is the induced metric ofdand µxY

is the restriction operator, namely, µxY(A) :=µ(Y^{n}∩A) for A⊂X^{n}.

Local to global: Let {Y_{i}^{n}}_{i∈I} be a finite open cover of a locally compact mm-space
(X^{n}, d, µ). Assume that (Y_{i}^{n}, d_{Y}_{i}) is a complete length space and (Y_{i}^{n}, d_{Y}_{i}, µxYi) satis-
fies Sc^{vol}^{n}(Y_{i}^{n}) ≥ κ ≥ 0, then (X^{n}, d, µ) satisfies Sc^{vol}^{n}(X^{n}) ≥ κ ≥ 0 and r_{X}^{n} can be
chosen as a partition of unity of the functions{r_{Y}^{n}

i }_{i∈I}.

Question 2.13. Assume that Sc^{vol}^{n}^{1}(X_{1}^{n}^{1}) ≥κ_{1}(≥ 0)for the compact mm-space (X_{1}^{n}^{1}, d_{1}, µ_{1})
and Sc^{vol}^{n}^{2} X_{2}^{n}^{2}

≥κ_{2}(≥0)for the compact mm-space (X_{2}^{n}^{2}, d_{2}, µ_{2}), then do we have
Sc^{vol}^{n}^{1+}^{n}^{2} X_{1}^{n}^{1} ×X_{2}^{n}^{2}

≥κ_{1}+κ_{2}, r_{X}^{n}

1×X_{2}^{n} = min{r_{X}^{n}

1, r_{X}^{n}

2}
for X_{1}^{n}^{1} ×X_{2}^{n}^{2}, d_{3}, µ_{3}

? Here X_{1}^{n}^{1} ×X_{2}^{n}^{2} is endowed with the measure µ_{3} :=µ_{1}⊗µ_{2} and with
the Pythagorean product metric d_{3}:=p

d^{2}_{1}+d^{2}_{2}.

### 3 smGH-convergence

Let {µ_{n}}_{n∈N} and µ be Borel measures on the space X, then the sequence {µ_{n}}_{n∈N} is said to
convergestrongly (also called setwise convergence in other literature ) to a limitµif lim

n→∞µn(A) = µ(A) for everyAin the Borel σ-algebra.

A map f: X → Y is called an -isometry between compact metric spaces X and Y, if

|d_{X}(a, b)−dY(f(a), f(b))| ≤ for alla, b ∈X and it is almost surjective, i.e., for every y ∈Y,
there exists an x∈X such thatd_{Y}(f(x), y)≤.

In fact, iff is an-isometry X→Y, then there is a (4)-isometry f^{0}:Y →X such that for
all x∈X,y∈Y,dX(f^{0}◦f(x), x)≤3,dY(f◦f^{0}(y), y)≤.

Definition 3.1 (smGH-convergence). Let (Xi, di, µi)i∈N and (X, d, µ) be compact mm-spaces.

X_{i} converges to X in the strongly measured Gromov–Hausdorff topology (smGH-convergence)
if there are measurable _{i}-isometriesf_{i}:X_{i} → X such that _{i} →0 and fi∗µ_{i} → µin the strong
topology of measures asi→ ∞.

If the spaces (X_{i}, d_{n}, µ_{i}, p_{i})i∈N and (X, d, µ, p) are locally compact pointed mm-spaces, it
is said that X_{i} converges to X in the pointed strongly measured Gromov–Hausdorff topolo-
gy (psmGH-convergence) if there are sequences ri → ∞, i → 0, and measurable pointed
_{i}-isometriesf_{i}:B_{r}_{i}(p_{i})→B_{r}_{i}(p), such thatfi∗µ_{i}→µ, where the convergence is strong conver-
gence.

Remark 3.2. Let (X_{i}, d_{i}, µ_{i})i∈Nconverge to (X, d, µ) in the measured Gromov–Hausdorff topo-
logy, then there are measurable_{i}-isometriesf_{i}:X_{i}→X such thatfi∗µ_{i} weakly converges toµ.

If there is a Borel measureν onX such that sup

i

fi∗µi ≤ν, i.e., sup

i

fi∗µi(A)≤ν(A) for every A in the Borel σ-algebra onX, then Xi smGH-converges toX (see [26, Lemma 4.1]).

Remark 3.3. Then-dimensional condition is not preserved by the measured Gromov–Hausdorff convergence as the following example shows. Let

aiS^{2}:= S^{2}, aid_{S} (ai ∈(0,1)) be a sequence
of space, then the limit of a_{i}S^{2} under the measured Gromov–Hausdorff convergence is a point
when ai goes to 0. The limit exists as the Ricci curvature ofaiS^{2} is bounded below by 1.

Remark 3.4. Then-dimensional condition is not preserved by the smGH-convergence since the limits of lim

r→0 lim

i→∞

µi(Br(x))

volE(Br(R^{n})) may not be commutative for some mm-spaces (X, d, µi). Assume
the total variation distance of the measures goes to 0 as i→ ∞, i.e.,

d_{T V}(µ_{i}, µ) := sup

A

|µ_{i}(A)−µ(A)| →0,

where Aruns over the Borelσ-algebra of X, then the limits are commutative.

One can also define the total variation Gromov–Hausdorff convergence (tvGH-convergence) for mm-spaces by replacing the strong topology with the topology induced by the total variation distance in definition of smGH-convergence. Then tvGH-convergence implies smGH-convergence and then-dimensional condition is preserved by tvGH-convergence.

Theorem 3.5 (stability). If compact mm-spaces (X_{i}^{n}, d_{i}, µ_{i}) with Sc^{vol}^{n}(X_{i}^{n}) ≥ κ ≥ 0, SC-
radius rX^{n}_{i} ≥R >0, and (X_{i}^{n}, di, µi) smGH-converge to the compact mm-space (X^{n}, d, µ) with
n-dimensional condition, then X^{n} also satisfies Sc^{vol}^{n}(X^{n})≥κ and the SC-radius r_{X}^{n} ≥R.

Proof . Fix anx∈X^{n} and letBr(x) be the small r-ball on X^{n}where r < R, then there exists
x_{i} ∈ X_{i}^{n} such that f_{i}^{−1}(B_{r}(x)) ⊂ B_{r+4}_{i}(x_{i}) where B_{r+4}_{i}(x_{i}) ⊂ X_{i}^{n} and r+ 4_{i} ≤ R. Thus,
fi∗µi(Br(x))≤µi(Br+4i(xi)).

For κ= 0, since Sc^{vol}^{n}(X_{i}^{n})≥0 and SC-radius≥R >0, then µ_{i}(B_{r}(x_{i}))≤vol_{E}(B_{r}(R^{n}))
for all 0 < r≤R and all i. Therefore, fi∗µi(Br(x))< µi(Br+4i(xi))≤volE(Br+4i(R^{n}))
forr+ 4i ≤R. Sincei that is not related tor can be arbitrarily small, thenµ(Br(x))≤
vol_{E}(B_{r}(R^{n})).

For κ > 0, we have µ_{i}(B_{r}(x_{i})) < volS×E B_{r} S^{2}(γ)×R^{n−2}

for all 0 < r ≤ R, all i, and γ >

q2

κ. Thus, fi∗µi(Br(x))< µi(Br+4i(xi))<volS×E Br+4i S^{2}(γ)×R^{n−2}
for
r+ 4_{i} ≤ R. Since _{i} that is not related to r can be arbitrarily small, then µ(B_{r}(x)) ≤
volS×E B_{r} S^{2}(γ)×R^{n−2}

for γ >

q2

κ. Thus, µ(B_{r}(x)) < volS×E B_{r} S^{2}
q 2

κ+^{0}

×
R^{n−2}

, where 0< ^{0} is independence on r and ^{0} can as small as we want. Therefore, we

have Sc^{vol}^{n}(X^{n})≥κ.

Definition 3.6 (tangent space). The mm-space (Y, dY, µY, o) is a tangent space of (X^{n}, d, µ)
atp∈X^{n}if there exists a sequence λ_{i} → ∞such that (X^{n}, λ_{i}·d, λ^{n}_{i} ·µ, p) psmGH-converges to
(Y, d_{Y}, µ_{Y}, o) as λ_{i} → ∞.

Therefore, (Y, d_{Y}, µ_{Y}, o) also satisfies then-dimensional condition and can be written asY^{n}.
Corollary 3.7. Assume the compact mm-space (X^{n}, d, µ) with Sc^{vol}^{n}(X^{n}) ≥ κ ≥ 0 and the
tangent space(Y^{n}, dY, µY, o)ofX^{n}exists at the pointp, then(Y^{n}, dY, µY, o)satisfiesSc^{vol}^{n}(Y^{n})

≥0 and the SC-radius≥r_{X}^{n}.

Proof . Since then-volumic scalar curvature has the quadratic scaling property, i.e., Sc^{vol}^{n}(λX^{n})

≥λ^{−2}κ≥0 andr_{λX}^{n} =λr_{X}^{n} for allλ >0, whereλX^{n}:= (X^{n}, λ·d, λ^{n}·µ), then Sc^{vol}^{n}(Y^{n})≥0

is implied by the stability theorem.

The mm-spaces with Sc^{vol}^{n} ≥0 includes some of the Finsler manifolds, for instance,R^{n}equip-
ped with any norm and with the Lebesgue measure satisfies Sc^{vol}^{n} ≥0 and any smooth compact
Finsler manifold is a CD(κ, n) space for appropriate finite κ and n [30]. It is well-known that
Gigli’s infinitesimally Hilbertian [12] can be seen as the Riemannian condition in RCD(κ, n)
space. Thus, infinitesimally Hilbertian can also be used as a Riemannian condition in the mm-
spaces with Sc^{vol}^{n} ≥0.

Definition 3.8 (RSC(κ, n) space). The compact mm-space (X^{n}, d, µ) with the n-dimensional
condition is a Riemannian n-volumic scalar curvature≥ κ space (RSC(κ, n) space) if it is in-
finitesimally Hilbertian and satisfies the Sc^{vol}^{n}(X^{n})≥κ≥0.

Note that any finite-dimensional Alexandrov spaces with curvature bounded below are in- finitesimally Hilbertian. Then

Al^{n}(κ)⇒RCD((n−1)κ, n)⇒RSC((n(n−1)κ, n)

on (X^{n}, d,H^{n}), where the measureH^{n}is then-dimensional Hausdorff measure that satisfies the
n-dimensional condition.

Question 3.9. Are RSC(κ, n) spaces stable under tvGH-convergence?

Remark 3.10 (convergence of compact mm-spaces). For the compact metric measure spaces with probability measures, one can consider mGH-convergence, Gromov–Prokhorov conver- gence, Gromov–Hausdorff–Prokhorov convergence, Gromov–Wasserstein convergence, Gromov–

Hausdorff–Wasserstein convergence, Gromov’s 2-convergence, Sturm’sD-convergence [39, Sec- tion 27], and Gromov–Hausdorff-vague convergence [3]. smGH-convergence implies those con- vergences for compact metric measure spaces with probability measures, since the measures converge strongly in smGH-convergence and converge weakly in other situations.

Note that mm-spaces with infinitesimally Hilbertian are not stable under mGH-convergen- ce [12]. It is not clear if the infinitesimally Hilbertian are preserved under smGH-convergence or tvGH-convergence.

### 4 Smooth mm-space with Sc

_{α,β}

### > 0

Let the smooth metric measure space M^{n}, g,e^{−f}dVolg

(also known as the weighted Rieman-
nian manifold in some references), wheref is aC^{2}-function onM^{n},gis aC^{2}-Riemannian metric
and n≥2, satisfy the curvature-dimension condition CD(κ, n) for κ≥0, thenM^{n}also satisfies
Sc^{vol}^{n}(M^{n})≥nκ.

Motivated by the importance of the Ricci Bakry-Emery curvature, i.e.,
Ric^{M}_{f} = Ricc + Hess(f),

the weighted sectional curvature of smooth mm-space was proposed and discussed in [40]. On
the other hand, Perelman defined and used the P-scalar curvature in his F-functional in [31,
Section 1]. Inspired by the P-scalar curvature, i.e., Scg+ 24_{g}f− k 5_{g}fk^{2}_{g}, we propose another
scalar curvature on the smooth mm-space.

Definition 4.1 (weighted scalar curvature Sc_{α,β}). The weighted scalar curvature Sc_{α,β} on the
smooth mm-space M^{n}, g,e^{−f}dVolg

is defined by
Sc_{α,β}:= Scg+α4_{g}f −βk 5_{g}fk^{2}_{g}.

Note that the Laplacian 4_{g} here is the trace of the Hessian and Sc^{vol}^{n}(M^{n}) ≥ κ ≥ 0
is equivalent to Scα,β ≥ κ ≥ 0 for α = 3 and β = 3 (see [33, Theorem 8] or the proof of
Corollary 4.10below).

Example 4.2.

1. For α = ^{2(n−1)}_{n} and β = ^{(n−1)(n−2)}_{n}2 , the Sc^{2(n−1)}

n ,^{(n−1)(n−2)}

n2 is the Chang–Gursky–Yang’s
conformally invariant scalar curvature for the smooth mm-space [7]. That means for aC^{2}-
smooth function w onM^{n}, one has

Sc2(n−1)

n ,^{(n−1)(n−2)}

n2

e^{2w}g

= e^{−2w}Sc2(n−1)

n ,^{(n−1)(n−2)}

n2

(g).

2. For α = 2 and β = ^{m+1}_{m} , where m ∈ N∪ {0,∞}, the Sc_{2,}m+1

m is Case’s weighted scalar curvature and Case also defined and studied the weighted Yamabe constants in [6]. Case’s weighted scalar curvature is the classical scalar curvature if m = 0. If m=∞, then it is Perelman’s P-scalar curvature.

Note that the results in this paper are new for those examples.

4.1 Spin manifold and Sc_{α,β} > 0

For an orientable closed surface with density Σ, g,e^{−f}dVol_{g}

with Sc_{α,β} >0 and β ≥0, then
the inequality,

0<

Z

Σ

Sc_{α,β}dVol_{g} =
Z

Σ

Sc_{g}+α4_{g}f −βk 5_{g}fk^{2}_{g}

dVol_{g}= 4πχ(Σ)−β
Z

Σ

k 5_{g}fk^{2}_{g}dVol_{g},
implies thatχ(Σ)>0. Thus, Σ is a 2-sphere.

The following proposition of vanishing harmonic spinors is owed to Perelman essentially and the proof is borrowed from [1, Proposition 1].

Proposition 4.3(vanishing harmonic spinors). Assume the smooth mm-space M^{n},g,e^{−f}dVolg

is closed and spin. If α∈R,β ≥ ^{|α|}_{4}^{2} andSc_{α,β}>0, then the harmonic spinor ofM^{n} vanishes.

Proof . Let ψ be a harmonic spinor, though the Schr¨odinger–Lichnerowicz–Weitzenboeck for- mula

D^{2} =5^{∗}5+1
4Scg,

one has 0 =

Z

M

k 5_{g}ψk^{2}_{g}+ 1

4 Sc_{α,β}−α4_{g}f+βk 5_{g}fk^{2}_{g}
kψk^{2}_{g}

dVol_{g}

= Z

M

k 5_{g}ψk^{2}_{g}+
1

4Scα,β+β

4k 5_{g}fk^{2}_{g}

kψk^{2}_{g}+ α
4

5_{g}f,5_{g}kψk^{2}_{g}

g

dVolg. Then one gets

|α|

4 |

5_{g}f,5_{g}kψk^{2}_{g}

g| ≤ |α|

4 ck 5_{g}fk_{g}kψk_{g}×2c^{−1}k 5_{g}ψk_{g}

≤ |α|c^{2}

8 k 5_{g}fk^{2}_{g}kψk^{2}_{g}+c^{−2}|α|

2 k 5_{g}ψk^{2}_{g}.
Therefore,

0≥ Z

M

1−c^{−2}|α|

2

k 5_{g}ψk^{2}_{g}+2β−c^{2}|α|

8 k 5_{g}fk^{2}_{g}kψk^{2}_{g}+1

4Sc_{α,β}kψk^{2}_{g}

dVol_{g},

where c6= 0. If c^{−2}|α| ≤ 2,β ≥ ^{c}^{2}_{2}^{|α|} and Sc_{α,β} >0, then ψ= 0. So the conditions α∈ Rand
β ≥ ^{|α|}_{4}^{2} are needed

|α|

4 |

5_{g}f,5_{g}kψk^{2}_{g}

g| ≤ |α|

4 k 5_{g}fk_{g}kψk_{g}×2k 5_{g}ψk_{g}

= |α|

2 c_{1}k 5_{g}fk_{g}kψk_{g}×c^{−1}_{1} k 5_{g}ψk_{g}

≤ |α|

4 c^{2}_{1}k 5_{g}fk^{2}_{g}kψk^{2}_{g}+c^{−2}_{1} k 5_{g}ψk^{2}_{g}
.

Thus, 0≥

Z

M

1−c^{−2}_{1} |α|

4

k 5_{g}ψk^{2}_{g}+β−c^{2}_{1}|α|

4 k 5_{g}fk^{2}_{g}kψk^{2}_{g}+ 1

4Sc_{α,β}kψk^{2}_{g}

dVol_{g},

where c_{1} 6= 0. If c^{−2}_{1} |α| ≤4, β ≥c^{2}_{1}|α| and Sc_{α,β} > 0, thenψ = 0. Also the conditions α ∈ R

and β ≥ ^{|α|}_{4}^{2} are needed.

The following 3 corollaries come from the proposition of vanishing of harmonic spinors.

Corollary 4.4. Assume the smooth mm-space M^{n}, g,e^{−f}dVol_{g}

is closed and spin. Ifα∈R,
β ≥ ^{|α|}_{4}^{2} and Scα,β >0, then the A-genus and the Rosenberg index ofb M^{n} vanish.

Proof . Since the C^{∗}(π1(M^{n}))-bundle in the construction of the Rosenberg index [35] is flat,
there are no correction terms due to curvature of the bundle. Then the Schr¨odinger–Lichnero-
wicz–Weitzenboeck formula and the argument in the proof of vanishing harmonic spinors can

be applied without change.

Corollary 4.5. Assume thatM^{n}is a closed spinn-manifold andf is a smooth function onM^{n}.
If one of the following conditions is met,

(1) N ⊂ M^{n} is a codimension one closed connected submanifold with trivial normal bundle,
the inclusion of fundamental groups π1 N^{n−1}

→ π1(M^{n}) is injective and the Rosenberg
index of N does not vanish, or

(2) N ⊂ M^{n} is a codimension two closed connected submanifold with trivial normal bundle,
π_{2}(M^{n}) = 0, the inclusion of fundamental groups π_{1} N^{n−1}

→ π_{1}(M^{n}) is injective and
the Rosenberg index of N does not vanish, or

(3) N =N1∩ · · · ∩N_{k}, where N1· · ·N_{k} ⊂M are closed submanifolds that intersect mutually
transversely and have trivial normal bundles. Suppose that the codimension of N_{i} is at
most two for all i∈ {1 ˙k} and π2(N)→π2(M) is surjective and A(Nˆ )6= 0,

then M^{n} does not admit a Riemnannian metric g such that the smooth mm-space M^{n}, g,
e^{−f}dvol_{g}

satisfiesSc_{α,β}>0 for the dimension n≥3,α∈R andβ ≥ ^{|α|}_{4}^{2}.

Proof . The results in the [22, Theorem 1.1] and [42, Theorem 1.9] can be applied to show that
the Rosenberg index of M^{n} does not vanish and Corollary 4.4implies the theorem.

Let R_{f}(M^{n}) := {(g, f)} be the space of densities, where g is a smooth Riemannian metric
on M^{n} and f is a smooth function onM^{n} andR^{+}_{f}(M^{n})⊂ R_{f}(M^{n}) is the subspace of densities
such that the smooth mm-space M^{n}, g,e^{−f}dvol_{g}

satisfies Sc_{α,β}>0. Furthermore, letR^{+}_{f}(M^{n})
be endowed with the smooth topology.

Corollary 4.6. Assume M^{n} is a closed spin n-manifold, n ≥ 3, α ∈ R and β ≥ ^{|α|}_{4}^{2} and
R^{+}_{f}(M^{n})6=∅, then there exists a homomorphism

Am−1: πm−1(R^{+}_{f}(M^{n}))→KOn+m

such that

A_{0} 6= 0, ifn≡0,1 (mod 8),

A1 6= 0, ifn≡ −1,0 (mod 8),

A8j+1−n6= 0, if n≥7 and8j−n≥0.

Proof . Since the results in the [23, Section 4.4] and [9] depend on the existence of exotic
spheres with non-vanishing α-invariant. Let φ: M^{n} → M^{n} be a diffeomorphism of M^{n} and
(g, f) ∈ R^{+}_{f}(M^{n}), then (φ^{∗}g, f ◦φ) is also in R^{+}_{f}(M^{n}). Combining it with Proposition 4.3
shows that Hitchin’s construction of the map A[23, Proposition 4.6] can be applied to the case
of R^{+}_{f}(M^{n}) and then we can finish the proof with the arguments in [23, Section 4.4] and [9,

Section 2.5].

4.2 Conformal to PSC-metrics

Proposition 4.7 (conformal to PSC-metrics). Let M^{n}, g,e^{−f}dVol_{g}

be a closed smooth mm-
space with Sc_{α,β}>0. If the dimension n≥3, α∈R andβ ≥ ^{(n−2)|α|}_{4(n−1)}^{2}, then there is a metric g˜
conformal to g with positive scalar curvature (PSC-metric).

Proof . One only needs to show for all nontrivial u, R

M−uL_{g}udVol_{g} > 0 as in the Yamabe
problem [36], where

Lg :=4_{g}− n−2
4(n−1)Scg

is conformal Laplacian operator. To see this, Z

M

−uL_{g}udVol_{g} =
Z

M

k 5_{g}uk^{2}_{g}+ n−2

4(n−1)Sc_{g}u^{2}
dVol_{g}

= Z

M

k 5_{g}uk^{2}_{g}+ n−2

4(n−1) Sc_{α,β}−α4_{g}f +βk 5_{g}fk^{2}_{g}
u^{2}

dVol_{g}

= Z

M

k 5_{g}uk^{2}_{g}+ n−2

4(n−1)(Scα,β+βk 5_{g}fk^{2}_{g})u^{2}
+α(n−2)

2(n−1)h5_{g}f,5_{g}ui_{g}u

dVolg. Through the inequality

h5_{g}f,5_{g}ui_{g}u≤c_{2}k 5_{g}fk_{g}u×c^{−1}_{2} k 5_{g}uk_{g} ≤ c^{2}_{2}k 5_{g}fk^{2}_{g}u^{2}+c^{−2}_{2} k 5_{g}uk^{2}_{g}

2 ,

one gets Z

M

−uL_{g}udVol_{g} ≥
Z

M

1−|α|c^{−2}_{2} (n−2)
4(n−1)

k 5_{g}uk^{2}_{g}

+ β− |α|c^{−2}_{2}

(n−2)

4(n−1) k 5_{g}fk^{2}_{g}u^{2}+ n−2

4(n−1)Scα,βu^{2}

dVolg,
where c_{2}6= 0.

If|α|c^{−2}_{2} ≤ ^{4(n−1)}_{n−2} ,β ≥c^{2}_{2}|α|and Scα,β>0, then
Z

M

−uL_{g}udVolg >0.

So the conditionsn >2,α∈Rand β ≥ ^{(n−2)α}_{4(n−1)}^{2} are needed.

Remark 4.8. The proof was borrowed from [1, Proposition 2]. The two propositions above offer
a geometric reason why the condition of the vanishing of A-genus (without simply connectedb
condition) does not imply that M^{n} can admit a PSC-metric for the closed spin manifold M^{n}.

The proposition of conformal to PSC-metrics has following 3 corollaries.

Corollary 4.9 (weighted spherical Lipschitz bounded). Let M^{n}, g,e^{−f}dVol_{g}

be a closed ori-
entable smooth mm-space with Sc_{α,β} ≥ κ > 0, 3 ≤ n ≤ 8, α ∈ R and β ≥ ^{(n−2)|α|}_{4(n−1)}^{2}, then
the Lipschitz constant of the continuous map φ from M^{n}, g,e^{−f}dVolg

to the sphere S^{n} with
standard metric of non-zero degrees has uniformly non-zero lower bounded.

Proof . There is a metric ˜gconformal to g with scalar curvature≥n(n−1) by the proposition
of conformal PSC-metrics. For the continuous map φfrom (M^{n},g) to˜ S^{n} of non-zero degrees,
the Lipschitz constant of φ is greater than a constant that depends only on the dimensions n
by Gromov’s spherical Lipschitz bounded theorem [19, Section 3]. Since the conformal function
has the positive upper bound by the compactness of the manifold, then the Lipschitz constant

has uniformly non-zero lower bounded.

Corollary 4.10. For the closed smooth mm-space M^{n}, g,e^{−f}dVol_{g}

(n≥3) with Sc^{vol}^{n}(M^{n})

> 0, there is a metric gˆ conformal to g with PSC-metric. In particular, the A-genus andb Rosenberg index vanish with additional spin condition.

For the closed orientable smooth mm-space M^{n}, g,e^{−f}dVol_{g}

(3≤n≤8)withSc^{vol}^{n}(M^{n})≥
κ > 0, then the Lipschitz constant of the continuous map φ from M^{n}, g,e^{−f}dVolg

to the
sphere S^{n} with standard metric of non-zero degrees has uniformly non-zero lower bounded.