The Measure Preserving Isometry Groups of Metric Measure Spaces
Bochner's theorem says that if is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso( ) is finite. In this article, we show that if ( , , ) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure...
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| description | Bochner's theorem says that if is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso( ) is finite. In this article, we show that if ( , , ) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure-preserving isometry group Iso( , , ) is finite. We also give an effective estimate on the order of the measure-preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Émery Ricci curvature, except for small portions.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 114, 14 pages
The Measure Preserving Isometry Groups
of Metric Measure Spaces
Yifan GUO †‡
† Beijing Institute of Mathematical Sciences and Applications, Beijing, P.R. China
E-mail: yifan guo@foxmail.com
‡ Department of Mathematics, University of California, Irvine, CA, USA
Received June 30, 2020, in final form November 02, 2020; Published online November 10, 2020
https://doi.org/10.3842/SIGMA.2020.114
Abstract. Bochner’s theorem says that if M is a compact Riemannian manifold with
negative Ricci curvature, then the isometry group Iso(M) is finite. In this article, we show
that if (X, d,m) is a compact metric measure space with synthetic negative Ricci curvature in
Sturm’s sense, then the measure preserving isometry group Iso(X, d,m) is finite. We also give
an effective estimate on the order of the measure preserving isometry group for a compact
weighted Riemannian manifold with negative Bakry–Émery Ricci curvature except for small
portions.
Key words: optimal transport; synthetic Ricci curvature; metric measure space; Bochner’s
theorem; measure preserving isometry
2020 Mathematics Subject Classification: 53C20; 53C21; 53C23
1 Introduction
Throughout this article, a metric measure space (X, d,m) means that (X, d) is a complete
separable metric space, and m is a Borel σ-finite measure on X, which is also finite on bounded
sets. We denote by Iso(X, d,m) the group of isometries of (X, d) which preserve the measure m.
We also denote by # Iso(X, d,m) the number of elements in Iso(X, d,m).
An example of metric measure space is a weighted Riemannian manifold (M, g,m), which
means that (M, g) is an n-dimensional Riemannian manifold, and m = e−vvolg where volg is
the volume element associated with g and v : M → R is a C3 function. In this case, X = M ,
d = dg is the intrinsic metric induced by g and m = e−vvolg. For N ∈ [n,∞) we define the
N -Bakry–Émery Ricci tensor on (M, g,m) by
RicN,m := Ric +∇2v − dv ⊗ dv
N − n
with the convention that when N = n, we require v = 0 so that Ricn,m = Ric. In the case of
N =∞, we define
Ric∞,m := Ric +∇2v.
Also, we define the Witten Laplacian
∆mu := ∆gu− 〈∇v,∇u〉g.
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 at https://www.emis.de/journals/SIGMA/Gromov.html
mailto:yifan_guo@foxmail.com
https://doi.org/10.3842/SIGMA.2020.114
https://www.emis.de/journals/SIGMA/Gromov.html
2 Y. Guo
1.1 Synthetic Ricci curvature bounds
The study of synthetic Ricci curvature lower bounds for metric measure spaces originates from
the works of Bakry–Émery [7], Lott–Villani [24] and Sturm [29, 30]. These are the so-called
CD(K,N) spaces (K ∈ R, 1 ≤ N ≤ ∞), spaces satisfying (K,N)-curvature-dimension condition.
To rule out non-Riemannian Finsler manifolds, Ambrosio–Gigli–Savaré [4] and Gigli [15] add an
infinitesimal Hilbertianity condition to CD(K,N) by introducing the class of RCD(K,N) spaces,
spaces satisfying (K,N)-Riemannian-curvature-dimension condition. Since then, many geomet-
ric and analytic results of Ricci curvature lower bounds have been established on CD(K,N) and
RCD(K,N) spaces. For example, Bishop–Gromov comparison theorem [30], Cheeger–Gromoll
splitting theorem [14], Li-Yau estimates for heat flow [6, 13, 35], sharp Sobolev inequality [26],
and Levy–Gromov isoperimetric inequality [9], etc.
It is natural to ask whether there is a synthetic notion of Ricci curvature upper bound. This
question for Ric∞,m is addressed by Sturm in his paper [31]. Sturm’s definition is inspired
by the fact that for any infinitesimally Hilbertian space (X, d,m) the condition RCD(K,∞) is
equivalent to the inequality
W (Htδx,Htδy) ≤ e−Ktd(x, y) (1.1)
for all x, y ∈ X and t > 0 (see, e.g., [4]). Here, W is the 2-Wasserstein distance on P2(X) and
Ht is the heat flow or the gradient flow of the entropy functional Entm, while δx is the Dirac
measure at x (see Section 2 for definitions). The basic idea is to replace “≤” by “≥” in (1.1).
Solving K from W (Htδx,Htδy) ≥ e−Ktd(x, y), we define the quantities
θ+(x, y) = − lim inf
t→0+
1
t
log(W (Htδx,Htδy)/d(x, y)),
θ∗(x) = lim sup
y,z→x
θ+(y, z).
If (M, g,m) is a weighted Riemannian manifold, Sturm [31] proved that
θ∗(x) = sup
{
Ric∞,m(ξ, ξ)/|ξ|2 : ξ ∈ TxM, ξ 6= 0
}
for all x ∈M (see Section 2 for more details). This motivates the following definition.
Definition 1.1 (Sturm [31]). We say that the metric measure space (X, d,m) has synthetic
Ricci curvature upper bound K if θ∗(x) ≤ K for all x ∈ X.
However, it should be noted that due to the theorems of Gao–Yau [12] and Lohkamp [23],
Ricci curvature upper bounds do not place any topological restriction on the manifold. More
precisely, for any integer n ≥ 3, there exist two positive constants Λ1 = Λ1(n), Λ2 = Λ2(n) such
that each manifold M of dimension n admits a complete metric g with −Λ1 < Ric(g) < −Λ2.
Neither is the set of Riemannian manifolds with Ricci curvature upper bound precompact
in Gromov–Hausdorff convergence, since the number of small balls in a large ball goes to ∞ as
Ric → −∞. Given such flexibility of Ricci curvature upper bound, it is not surprising to learn
that there are not as many geometric results of Ricci curvature upper bounds as Ricci curvature
lower bounds. In fact, it was pointed out by Gromov in [16, Section 5] that the only result
widely known is Bochner’s theorem, which is the main subject of this article.
1.2 Bochner’s theorem
One of the classical results of a compact manifold with negative Ricci curvature (Ricci curvature
bounded above by 0) is Bochner’s theorem which states that the isometry group of the manifold
must be finite (see, e.g., [8]). In fact, Bochner showed that there exists no continuous group of
The Measure Preserving Isometry Groups of Metric Measure Spaces 3
isometry by considering the Laplacian of the square norm of the Killing vector field. This is the
genesis of the famous “Bochner technique” which produces numerous results.
Later, several authors tried to extend Bochner’s theorem by estimating the order of the
isometry group by various quantities. Before we mention them, we may see from the work of
Lohkamp that it is impossible to control the order of the isometry group merely in terms of
dimension and Ricci curvature bounds.
Theorem 1.2 (Lohkamp [23]). Let M be a compact n-dimensional manifold with n ≥ 3 and G
be a subgroup of Diff(M), the group of diffeomorphisms of M . Then G is the isometry group
of M for some metric g with Ric(g) < 0 if and only if G is finite.
Fifty years before Bochner’s result came out, Hurwitz [19] showed that when X is a Riemann
surface of genus g ≥ 2, the order of the automorphism group of X, # Aut(X) ≤ 84(g − 1).
Later, the estimate of the order of the isometry group was generalized to hyperbolic manifolds
by Huber [18], to manifolds with sectional curvature bounded above from 0 by Im Hof [20],
to manifolds with non-positive sectional curvature and Ricci curvature negative at some point
by Maeda [25] and to manifolds with non-positive sectional curvature and finite volume by
Yamaguchi [34]. For general compact Riemannian manifolds with negative Ricci curvature,
Katsuda [21] estimated the order of the isometry group by sectional curvature, dimension, di-
ameter and injectivity radius. In [10], Dai–Shen–Wei estimated the order of the isometry group
by Ricci bounds, dimension, volume, and injectivity radius. Recently, Katsuda–Kobayashi [22]
gave a bound of the order of the isometry group for manifolds with negative Ricci curvature
except for small portions.
There are also other generalizations of Bochner’s theorem. In [27], Rong showed that compact
manifolds with negative Ricci curvature do not admit non-trivial invariant F -structure which
includes the Killing vector field as a special case. Bagaev and Zhukova [5] extended Bochner’s
theorem to Riemannian orbifolds with negative Ricci curvature. From the works of Deng–
Hou [11] and Zhong–Zhong [36], we know that a compact Finsler manifold with negative Ricci
curvature has a finite isometry group. Van Limbeek [32] estimated the order of the isometry
group for manifolds on which circle actions do not exist. The list above is far from complete
and can go on and on.
Our first main result is a generalization of Bochner’s theorem to compact metric measure
spaces with synthetic negative Ricci curvature.
Theorem 1.3. Let (X, d,m) be a compact metric measure space with θ∗(x) < 0 for all x ∈ X.
Then Iso(X, d,m) is finite.
As a corollary, we have Bochner’s theorem for weighted Riemannian manifolds:
Corollary 1.4. Let (M, g,m) be a closed weighted Riemannian manifold with ∞-Bakry–Émery
Ricci tensor Ric∞,m < 0, then Iso(M, g,m) is finite.
This corollary can also be obtained by considering the Laplacian of the square norm of
divergence free Killing vector field on (M, g,m). In fact, with this method, we actually have
a stronger theorem.
Theorem 1.5. Let (M, g,m) be a closed n-dimensional weighted Riemannian manifold. If the
N -Bakry–Émery Ricci tensor RicN,m < 0 for some n ≤ N ≤ ∞, then Iso(M, g,m) is finite.
In view of this theorem, it should be expected that if there is a notion of synthetic N -
Ricci curvature upper bound in the future, then Bochner’s theorem should hold on spaces with
synthetic negative N -Ricci curvature.
Our next result estimates # Iso(M, g,m) by sectional curvature and other geometric quan-
tities for a weighted Riemannian manifolds M with negative ∞-Bakry–Émery Ricci curvature
4 Y. Guo
except for small portions. This is a generalization of the main theorem in [21] and Theorem 0.2
in [22]. We denote the sectional curvature by KM , the injectivity radius by injM , and the
Lp(M,m) norm by ‖ · ‖p, i.e., for any Borel function f : M → R
‖f‖p =
(∫
M
|f(x)|p dm(x)
)1/p
.
Theorem 1.6. Let (M, g,m) be a closed n-dimensional weighted Riemannian manifold satis-
fying
|KM | ≤ Λ1, |∇Ric∞,m | ≤ Λ2, m(M) ≤ V, diam(M) ≤ D, injM ≥ i0,
N ≥ n ≥ 3, RicN,m ≥ −Λ3
for some fixed positive constants i0, N , Λ1, Λ2, Λ3, V , D. If for some w > 0,
‖(θ∗ + w)+‖n/2 < min
(
1
4A
,
w
4B
)
, (1.2)
where (θ∗ + w)+ = max{θ∗ + w, 0}, and A, B are constants such that the Sobolev inequality
‖f‖22n/(n−2) ≤ A‖Df‖
2
2 +B‖f‖22 (1.3)
holds for all f ∈W 1,2(M, g,m), then there exists a constant L1 = L1(n, i0, N,Λ1,Λ2,Λ3, V,D,w,
A,B) such that # Iso(M, g,m) ≤ L1.
Condition (1.2) is saying that the Bakry–Émery Ricci curvature greater than −w has small
Ln/2 norm which is what we mean by “manifolds with negative Bakry–Émery Ricci curvature
except for small portions”.
Our final result generalizes Theorem 1.3 in [10] and Theorem 0.3 in [22]. It replaces the
sectional curvature in Theorem 1.6 by Ricci curvature.
Theorem 1.7. Let
(
M, g,m = e−vvolg
)
be a closed n-dimensional weighted Riemannian mani-
fold satisfying
n ≥ 3, |Ric | ≤ Λ1, |∇Ric∞,m | ≤ Λ2, ‖e−v‖∞ ≤ E,
diam(M) ≤ D, injM ≥ i0
for some fixed positive constants i0, Λ1, Λ2, E, D. If for some w > 0,
‖(θ∗ + w)+‖n/2 < min
(
1
2A
,
w
2B
)
,
where A,B are constants such that the Sobolev inequality (1.3) holds, then there exists a constant
L2 = L2(n, i0,Λ1,Λ2, E,D,w,A,B) such that # Iso(M, g,m) ≤ L2.
Remark 1.8. Theorem 1.6 and Theorem 1.7 are very similar but their proofs are different. In
the proof of Theorem 1.6, we use optimal transport to prove a key lemma (Lemma 4.1) which
is an analogue of Lemma 4.2 in [10] and Proposition 2.2 in [22]. On the contrary, the proof of
Theorem 1.7 is totally differential geometric. We have essentially used Lemma 4.2 in [10] and
Proposition 4.1 in [22] which is obtained by estimating the Jacobi fields. It would be interesting
to know whether there is an estimate of the order of the measure preserving isometry group on
metric measure space instead of weighted Riemannian manifold and we believe that the proof
of Theorem 1.6 gives more insight in this direction.
The Measure Preserving Isometry Groups of Metric Measure Spaces 5
Remark 1.9. It should also be noted that the constant L1 in Theorem 1.6 is computable while
the constant L2 in Theorem 1.7 is not because some compactness argument is used in Lemma 4.2
of [10] on which Theorem 1.7 is relied.
The plan of the rest of the paper is as follows. In Section 2, we introduce synthetic Ricci
curvature upper bounds. In Section 3, we prove our generalization of Bochner’s theorem to
metric measure spaces. In Section 4, we give our estimates on the order of the measure preserving
isometry groups.
2 Synthetic Ricci curvature upper bounds
Let (X, d,m) be a metric measure space such that
∫
X e−Cd
2(x0,x) dm(x) < ∞ for some x0 ∈ X
and C > 0. We denote by P (X) the set of all Borel probability measures on X. We define the
2-Wasserstein distance on P2(X) =
{
µ ∈ P (X) :
∫
X d
2(x0, x) dµ(x) <∞ for some x0 ∈ X
}
by
W (µ, ν) =
(
inf
π
∫
X×X
d2(x, y) dπ(x, y)
)1/2
,
where the infimum is taken among all π ∈ P (X×X) such that π has marginals µ and ν. Such π
is called an admissible plan. The measure π at which the infimum is realized is called an optimal
transport plan of µ and ν, which always exists (see [33, Chapter 4]).
We define the entropy functional Entm : P2(X)→ R ∪ {+∞} by
Entm(µ) =
∫
X
ρ log ρdm, µ = ρm,
+∞, otherwise.
Let K ∈ R, N > 1 be two numbers. For t ∈ [0, 1], we define the functions β
(K,N)
t (x, y) on
X ×X by
β
(K,N)
t (x, y) =
+∞ if K > 0 and α > π,(
sin(tα)
t sinα
)N−1
if K > 0 and α ∈ [0, π],
1 if K = 0,(
sinh(tα)
t sinhα
)N−1
if K < 0,
where α =
√
|K|
N−1d(x, y).
Definition 2.1. We say that a metric measure space (X, d,m) satisfies curvature dimension
condition CD(K,∞) if for any two measures µ0 and µ1 in P2(X), there exists some geodesic (µt)
in P2(X) such that for all 0 ≤ t ≤ 1, we have
Entm(µt) ≤ (1− t) Entm(µ0) + tEntm(µ1)−
K
2
t(1− t)W 2(µ0, µ1).
A metric measure space (X, d,m) is said to satisfy curvature dimension condition CD(K,N) for
1 ≤ N <∞ if for any two measures µ0 and µ1 in P2(X), there exists some geodesic (µt = ρtm)
in P2(X) such that for some optimal transport plan π of µ0 and µ1 and all 0 ≤ t ≤ 1, we have∫
X
ρ
1− 1
N
t (x) dm(x) ≥ (1− t)
∫
X×X
ρ0(x0)
− 1
N β1−t(x0, x1)
1
N dπ(x0, x1)
+ t
∫
X×X
ρ1(x1)
− 1
N βt(x0, x1)
1
N dπ(x0, x1).
6 Y. Guo
In the case of n-dimensional weighted Riemannian manifold, CD(K,∞) is equivalent to
Ric∞,m ≥ K, and CD(K,N) is equivalent to N ≥ n, RicN,m ≥ K (see, e.g., [24, 29, 30, 33]).
Let (X, d,m) be a metric measure space and f ∈ L2(X,m). We define the Cheeger energy
of f by
Ch(f) = inf
{
lim inf
n→∞
1
2
∫
X
|Dfn|2dm : ‖fn − f‖L2(X,m) → 0, fn ∈ Lip(X, d)
}
,
where for any function g : X → R, the slope |Dg| is defined as
|Dg|(x) = lim sup
y→x
|g(y)− g(x)|
d(y, x)
.
We also define the descending slope |D−g| by
|D−g|(x) = lim sup
y→x
[g(y)− g(x)]−
d(y, x)
.
Being a convex lower semicontinuous function, the Cheeger energy admits a gradient flow Ht on
L2(X,m).
However, the Cheeger energy Ch is not neccessarily a quadratic form. We say that (X, d,m)
is infinitesimally Hilbertian if Ch is a quadratic form, i.e.,
Ch(f + g) + Ch(f − g) = 2Ch(f) + 2Ch(g) for every f, g ∈ L2(X,m).
We also define the metric gradient flow of a function E : X → R ∪ {+∞} to be a locally
absolutely continuous curve (µt) on X such that for all t ≥ 0 we have
E(µ0) = E(µt) +
1
2
∫ t
0
|µ′s|2 ds+
1
2
∫ t
0
|D−E|2(µs) ds, (2.1)
where |µ′t| = lim
s→t
d(µs,µt)
|s−t| . We denote by Ht the metric gradient flow of Entm on P2(X). For
more details on metric gradient flow, see, e.g., [2].
Definition 2.2 (Ambrosio–Gigli–Modino–Rajala [1], Ambrosio–Gigli–Savaré [4], Gigli [15]).
Let K ∈ R and 1 < N ≤ ∞. A metric measure space (X, d,m) is said to have N -Riemannian
Ricci curvature bounded below by K, or to satisfy RCD(K,N) condition, if (X, d,m) satisfies
CD(K,N) and Ch is a quadratic form on L2(X,m).
In RCD(K,∞) spaces, heat flow behaves well. It is the result of Ambrosio–Gigli–Savaré [3, 4]
that the gradient flow Ht of Ch and the gradient flow Ht of Entm coincides and that for every
f ∈ L2(X,m) ∩ L∞(X,m), and every x ∈ X we have
Htf(x) =
∫
X
f(z) dHtδx(z). (2.2)
Moreover, the heat flow is a EVIK gradient flow so that it satisfies the contraction property (1.1).
Turning to the Ricci curvature upper bounds, we recall the definition of
θ+(x, y) = − lim inf
t→0+
1
t
log(W (Htδx,Htδy)/d(x, y)).
For a function u : R → R, we write ∂−t u := lim inf
s→t
(u(s) − u(t))/(s − t). Thus in this nota-
tion, θ+(x, y) = − ∂−t
∣∣
t=0
logW (Htδx,Htδy). Let γ : [a, b] → M be a geodesic in a weighted
Riemannian manifold (M, g,m). We define the average ∞-Ricci curvature along γ by
ρ(γ) =
1
b− a
∫ b
a
Ric∞,m (γ̇(t), γ̇(t)) / |γ̇(t)|2 dt.
Sturm has proved the following two sided bounds for θ+(x, y).
The Measure Preserving Isometry Groups of Metric Measure Spaces 7
Theorem 2.3 (Sturm [31]). Let (M, g,m) be an n-dimensional weighted Riemannian manifold
satisfying RCD(−K ′, N) condition for some finite K ′ ≥ 0, N > 1. Let x, y be non-conjugate
points in M , and γ : [0, T ]→M be a unit speed geodesic connecting the two points. Then we
have
ρ(γ) ≤ θ+(x, y) ≤ ρ(γ) + σ(γ) · tan2
(√
σ(γ)d(x, y)/2
)
, (2.3)
where d is the metric induced by g, and
σ(γ) = max
t∈[0,T ]
n∑
i,j=1
|R(ei, γ̇, ej , γ̇)|2
1
2
for R the Riemann curvature tensor and {ei}ni=1 some parallel orthonormal frame along γ such
that e1 = γ̇.
Taking the endpoints x, y infinitely close to each other in (2.3), we have
Corollary 2.4. Let (M, g,m) be a weighted Riemannian manifold satisfying RCD(−K ′, N) con-
dition for some finite K ′ ≥ 0, N > 1. Then we have
θ∗(x) = sup
{
Ric∞,m(ξ, ξ)/|ξ|2 : ξ ∈ TxM, ξ 6= 0
}
.
In view of the corollary, we may view θ∗(x) as a counterpart of Ric∞,m in metric measure
space which justifies the Definition 1.1.
3 Generalization of Bochner’s theorem
to metric measure spaces
In this section we will prove our first generalization of Bochner’s theorem. A fundamental lemma
is the following discrete version of Bochner’s theorem which is a standard technique (cf. [21]).
For any Borel map f : X → Y between metric spaces, µ a Borel measure on X, we denote
by f#µ the Borel measure on Y defined by f#µ(A) = µ
(
f−1(A)
)
for all A ⊂ Y Borel.
Lemma 3.1. Let (X, d,m) be a compact metric measure space with θ∗(x) < 0 for all x ∈ X.
Then there exists λ > 0 such that if φ ∈ Iso(X, d,m) satisfies d(x, φ(x)) ≤ λ for all x ∈ X,
then φ is the identity map.
Proof. Since for any fixed x ∈ X, θ∗(x) = lim sup
y,z→x
θ+(y, z) < 0, there exist ε = ε(x) > 0
and λx > 0 such that for all y, z ∈ Bλx(x), we have θ+(y, z) < −ε. Since X =
⋃
xBλx/2(x),
by compactness, there exist x1, . . . , xN ∈ X such that X =
⋃N
i=1Bλxi/2(xi). Now let λ =
min{λxi/2: 1 ≤ i ≤ N}. We claim that this λ is what we want.
Let y ∈ X and z = φ(y) be such that d(y, z) = max
x∈X
d(x, φ(x)) ≤ λ. There exists 1 ≤ i ≤ N ,
such that y ∈ Bλxi/2(xi). Since d(y, z) ≤ λ, we have d(xi, z) ≤ d(xi, y) + d(y, z) < λxi . Thus
y, z ∈ Bλxi (xi) for some 1 ≤ i ≤ N . Therefore, θ+(y, z) = −∂−t |t=0 log(W (Htδy,Htδz)) < −ε.
Hence for t > 0 small, we have
W (Htδy,Htδz) ≥ eεt/2d(y, z). (3.1)
On the other hand, since φ is an isometry that preserves the measure m, we have W (φ#µ, φ#ν) =
W (µ, ν), Entm(φ#µ) = Entm(µ), |(φ#µt)′| = |µ′t|, and |D− Entm |(φ#µ) = |D− Entm |(µ). In
view of the definition (2.1), we see that (φ#µt) is a gradient flow of Entm if (µt) is. Hence,
8 Y. Guo
φ#(Htδy) = Ht(φ#δy) = Htδφ(y) = Htδz. Therefore, (id × φ)#(Htδy) is an admissible plan of
(Htδy,Htδz). So we get for all t > 0
W 2(Htδy,Htδz) ≤
∫
X
d2(x, φ(x)) dHtδy(x) ≤ max
x∈X
d2(x, φ(x)) = d2(y, z). (3.2)
Combining (3.1) and (3.2), we get, for t > 0 small, d2(y, z) ≥ eεtd2(y, z), which is impossible
unless d(y, z) = 0, i.e., φ is the identity map. �
Now, we can prove our first theorem. The proof is exactly the same as the second part of the
proof of [21] which is strongly inspired by [25].
Proof of Theorem 1.3. Take λ as in Lemma 3.1 and a = λ/4. By compactness, let x1, . . . , xN
∈ X be such that X =
⋃N
i=1Ba(xi). We define a map F from the measure preserving isometry
group Iso(X, d,m) to the symmetric group SN of degree N by F (φ) : i→ j(i) where j(i) is the
smallest j such that φ(xi) ∈ Ba(xj). We claim that F is injective. Assume F (φ) = F (ψ) = j(·).
For any x ∈ X, we have x ∈ Ba(xi) for some i, then
d(φ(x), ψ(x)) ≤ d(φ(x), φ(xi)) + d(φ(xi), xj(i)) + d(xj(i), ψ(xi)) + d(ψ(xi), ψ(x))
≤ 4a = λ.
By Lemma 3.1, φ = ψ. Hence F is injective and we get
# Iso(X, d,m) ≤ #SN = N !. (3.3)
This finishes the proof. �
Next we prove Theorem 1.5.
Proof of Theorem 1.5. Being a closed subgroup of the isometry group Iso(M, g), the mea-
sure preserving isometry group Iso(M, g,m) is a compact Lie group. (In general, the measure
preserving isometry groups of RCD(K,N) spaces are Lie groups. See [17, 28].) So it suffices to
show that its Lie algebra is of dimension 0. Let ξ be a vector field on M such that the flow φt
of ξ preserves g and m, i.e., (φt)
∗g = g, (φt)#m = m. Taking Lie derivatives, we have for all
vector fields U,W on M
〈∇Uξ,W 〉+ 〈∇W ξ, U〉 = 0, (3.4)
divm ξ = divg ξ − 〈∇V, ξ〉 = 0, (3.5)
where 〈 , 〉 denotes g and divg is the divergence operator associated with g. By (3.4), ∇ξ is
antisymmetric, so divg ξ = tr(∇ξ) = 0, which by (3.5) implies 〈∇V, ξ〉 = 0. Hence we obtain
1
2
〈
∇V,∇|ξ|2
〉
= 〈∇∇V ξ, ξ〉 = −〈∇ξξ,∇V 〉 = −∇ξ〈ξ,∇V 〉+ 〈∇ξ∇V, ξ〉 = ∇2V (ξ, ξ).
By Bochner’s formula for Killing vector field (see, e.g., [8, Lemmas 2 and 3]), we have
1
2
∆g|ξ|2 = |∇ξ|2 − Ric(ξ, ξ).
Hence we get
1
2
∆m|ξ|2 = |∇ξ|2 − Ric∞,m(ξ, ξ).
The Measure Preserving Isometry Groups of Metric Measure Spaces 9
Noticing that dV ⊗ dV (ξ, ξ) = 〈∇V, ξ〉2 = 0, we have, for all n ≤ N ≤ ∞,
1
2
∆m|ξ|2 = |∇ξ|2 − RicN,m(ξ, ξ).
Integrating the above equality over M , we obtain∫
M
RicN,m(ξ, ξ) dm =
∫
M
|∇ξ|2 dm ≥ 0.
Since RicN,m < 0, we get ξ = 0. �
4 Estimating the order of the measure
preserving isometry group
In this section, we give an explicit dependence of the upper bound in (3.3). We will be discussing
in the context of a compact weighted Riemannian manifold (M, g,m). First we introduce some
notations.
For any continuous map φ : M → M , we set dφ(x) = d(x, φ(x)) and δφ = max
x∈M
d(x, φ(x)). If
φ : M →M is such a map that for all x ∈M , x and φ(x) are connected by a unique minimizing
geodesic γ (this is always true if δφ < injM ), then we define
ρφ(x) =
1
dφ(x)
∫ dφ(x)
0
θ∗(γ(t)) dt.
Lemma 4.1. Let (M, g,m) be a closed n-dimensional weighted Riemannian manifold with in-
jectivity radius injM ≥ i0 and sectional curvature |KM | ≤ Λ. Let φ : M → M be a mea-
sure preserving isometry. Then there exist positive constants δ0 = δ0(i0,Λ, n), C1 = C1(Λ, n),
C2 = C2(Λ, n), such that if δφ ≤ δ then
∆md
2
φ ≥ −2d2φρφ − C1 tan2(C2dφ)d2φ. (4.1)
Proof. Let φ : M → M be a measure preserving isometry. With the same argument as (3.2),
we have
W 2(Htδx,Htδφ(x)) ≤
∫
X
d2(z, φ(z)) dHtδx(z) = Htd
2
φ(x),
where we have used the formula (2.2) in the last equality. Hence for t > 0, we have
W 2(Htδx,Htδφ(x))− d2φ(x)
t
≤
Htd
2
φ(x)− d2φ(x)
t
.
Since
lim
t→0
W 2(Htδx,Htδφ(x)) = d2φ(x) and lim
t→0
Htd
2
φ(x) = d2φ(x),
we obtain
∂−t
∣∣
t=0
W 2(Htδx,Htδφ(x)) ≤ ∂−t
∣∣
t=0
Htd
2
φ(x).
Thus
2dφ(x) ∂−t
∣∣
t=0
W (Htδx,Htδφ(x)) ≤ ∆md
2
φ(x).
10 Y. Guo
By definition, we have
θ+(x, φ(x)) = −
∂−t
∣∣
t=0
W (Htδx,Htδφ(x))
d(x, φ(x))
,
so we get
−2d2φ(x)θ+(x, φ(x)) = 2dφ(x) ∂−t
∣∣
t=0
W (Htδx,Htδφ(x)) ≤ ∆md
2
φ(x). (4.2)
Since M is compact, it automatically satisfies some RCD(−K ′, N) condition. Choosing δ0 ≤ i0
2
and applying the upper bound estimate in (2.3) to x, φ(x), we have
θ+(x, φ(x)) ≤ ρφ(x) + σ(γ) · tan2
(√
σ(γ)dφ(x)/2
)
.
Since |KM | ≤ Λ, we have |R(X,Y,X, Y )| ≤ Λ for allX ⊥ Y and |X| = |Y | = 1. Let {ei}ni=1 be an
orthonormal frame with e1 = γ̇. Then we have |R(ei, ej , ei, ej)| ≤ Λ and
∣∣R((ei+ej)/
√
2, ek, (ei+
ej)/
√
2, ek)
∣∣ ≤ Λ for distinct i, j, k. It follows that |R(ei, γ̇, ej , γ̇)| ≤ 2Λ. Observing that
R(ei, γ̇, ej , γ̇) = 0 if i = 1 or j = 1, we have
σ(γ) = max
t∈[0,T ]
n∑
i,j=1
|R(ei, γ̇, ej , γ̇)|2
1
2
= max
t∈[0,T ]
n∑
i,j=2
|R(ei, γ̇, ej , γ̇)|2
1
2
≤
(
(n− 1)24Λ2
) 1
2 = 2(n− 1)Λ.
Therefore if δφ ≤ δ0 = min
{
i0
2 ,
π
2C2
}
where C2 =
√
(n− 1)Λ/2, we have
θ+(x, φ(x)) ≤ ρφ(x) + 2(n− 1)Λ tan2
(
dφ
√
(n− 1)Λ/2
)
.
Combining (4.2), we get
∆md
2
φ ≥ −2d2φρφ − C1 tan2(C2dφ)d2φ,
where C1 = 4(n− 1)Λ. �
The following lemma is a quantitative version of Lemma 3.1.
Lemma 4.2. Let (M, g,m) be a closed weighted Riemannian manifold of dimension n ≥ 3 with
|KM | ≤ Λ1, |∇Ric∞,m | ≤ Λ2, m(M) ≤ V, injM ≥ i0.
If for some w > 0
‖(θ∗ + w)+‖n/2 < min
(
1
4A
,
w
4B
)
,
where A, B are constants such that the Sobolev inequality (1.3) holds, then there exists a constant
δ = δ(n, i0,Λ1,Λ2, V,D,w,A,B) such that if δφ ≤ δ then φ = id.
Proof. The argument we use here is very close to the proof of [22, Theorem 2.1]. First suppose
δφ ≤ δ ≤ min
{
δ0,
1
C2
arctan
√
w
2C1
}
, where δ0, C1, C2 are as in Lemma 4.1 so that (4.1) holds
and C1 tan2(C2δφ) ≤ w
2 . Multiplying both sides of (4.1) by d2φ and integrating over M , we get
0 ≥
∫
M
∣∣Dd2φ∣∣2dm− ∫
M
(
2ρφd
4
φ + C1 tan2(C2dφ)d4φ
)
dm
The Measure Preserving Isometry Groups of Metric Measure Spaces 11
=
∥∥Dd2φ∥∥22 − 2
∫
M
(ρφ + w)d4φdm+
∫
M
(
2w − C1 tan2(C2dφ)
)
d4φdm
≥
∥∥Dd2φ∥∥22 − 2‖(ρφ + w)+‖n/2
∥∥d2φ∥∥22n/(n−2) +
(
2w − C1 tan2(C2δφ)
)∥∥d2φ∥∥22
≥
∥∥Dd2φ∥∥22 − 2‖(ρφ + w)+‖n/2
∥∥d2φ∥∥22n/(n−2) +
3
2
w
∥∥d2φ∥∥22
≥
∥∥Dd2φ∥∥22 − 2‖(ρφ + w)+‖n/2
(
A
∥∥Dd2φ∥∥22 +B
∥∥d2φ∥∥22)+
3
2
w
∥∥d2φ∥∥22,
where in the last inequality we have used Sobolev inequality (1.3) with f = d2φ.
We estimate the term ‖(ρφ + w)+‖n/2 by
‖(ρφ + w)+‖n/2 ≤
∥∥∥∥∥ 1
dφ(x)
∫ dφ(x)
0
(θ∗(γ(t))− θ∗(x))+dt
∥∥∥∥∥
n/2
+ ‖(θ∗ + w)+‖n/2
≤ δφΛ2V
2/n + ‖(θ∗ + w)+‖n/2. (4.3)
To get the last inequality, let t ∈ [0, dφ(x)] be fixed. By Corollary 2.4, we have θ∗(γ(t)) =
sup
{
Ric∞,m(ξ, ξ)/|ξ|2 : ξ ∈ Tγ(t)M, ξ 6= 0
}
. Let ξ ∈ Tγ(t)M , |ξ| = 1 be such that θ∗(γ(t)) =
Ric∞,m(ξ, ξ), and ξ(s), s ∈ [0, t] be a parallel vector field along γ such that ξ(t) = ξ. Since
|ξ(0)| = 1, θ∗(x) ≥ Ric∞,m(ξ(0), ξ(0)). Hence
(θ∗(γ(t))− θ∗(x))+ ≤ (Ric∞,m(ξ(t), ξ(t))− Ric∞,m(ξ(0), ξ(0)))+ ≤ t|∇Ric∞,m |
≤ Λ2t ≤ Λ2δφ.
Therefore,
0 ≥
[
1− 2A
(
δφΛ2V
2/n + ‖(θ∗ + w)+‖n/2
)] ∥∥Dd2φ∥∥22
+
[
3
2
w − 2B
(
δφΛ2V
2/n + ‖(θ∗ + w)+‖n/2
)] ∥∥d2φ∥∥22
>
(
1
2
− 2AδφΛ2V
2/n
)∥∥Dd2φ∥∥22 +
(
w − 2BδφΛ2V
2/n
)∥∥d2φ∥∥22.
If we require further that
δφ ≤ δ = min
{
δ0,
1
C2
arctan
√
w
2C1
,
1
4AΛ2V 2/n
,
w
2BΛ2V 2/n
}
then dφ = 0, so we get φ = id. �
Proof of Theorem 1.6. Since RicN,m ≥ −Λ3 for some Λ3 > 0 and N ≥ n ≥ 3, by gen-
eralized Bishop–Gromov theorem for CD(K,N) spaces (see, e.g., [30, Theorem 2.3] or [33,
Theorem 30.11]) the number of δ-balls contained in M is bounded above by a constant L =
L(D, δ,Λ3, N). Hence by the same argument as the proof of Theorem 1.3 we get
# Iso(M, g,m) ≤ L! =: L1. �
Next we are going to prove Theorem 1.7. We have the following analogue of Lemma 4.1.
Lemma 4.3. Let ε > 0 be arbitrary and
(
M, g,m = e−vvolg
)
be a closed weighted Riemannian
manifold of dimension n ≥ 3 with |Ric | ≤ Λ, ‖e−v‖∞ ≤ E, diam(M) ≤ D and injM ≥ i0. Then
there exists δ1 = δ1(ε, n,Λ, E,D, i0) such that if φ : M → M is a measure preserving isometry
of M with δφ ≤ δ then we have, in the sense of distribution
∆mdφ ≥ −dφρφ − εGdφ, (4.4)
where G is a function on M with a uniform Lp(M,m) bound ‖G‖p ≤ C(p, n,Λ, E,D, i0) inde-
pendent of ε for all 1 ≤ p <∞.
12 Y. Guo
This lemma is a consequence of Lemma 4.2 in [10], which we cite here:
Lemma 4.4 (Dai–Shen–Wei [10]). Let ε > 0 be arbitrary and M be a Riemannian manifold
satisfying |Ric | ≤ Λ, injM ≥ i0, and vol(M) ≤ V . Then there exists a δ2 = δ2(ε, n,Λ, i0, V )
such that for any isometry φ : M →M with δφ ≤ δ, we have, in the sense of distribution
∆gdφ ≥ −
∫ dφ
0
Ric(γ̇) dt− εGdφ, (4.5)
where γ is the unique minimizing geodesic connecting x and φ(x) and G is a function on M
with a uniform Lp(M, vol) bound
‖G‖Lp(M,vol) ≤ Cp(n,Λ, i0, V )
independent of ε for all 1 ≤ p <∞. Moreover, inequality (4.5) is pointwise for all x ∈ M such
that x 6= φ(x).
Remark 4.5. The statement of Lemma 4.4 may seem stronger than Lemma 4.2 of [10], but
a close inspection on the proof of Lemma 4.2 of [10] shows that it actually proves the fact we
state here. This fact is also used in Proposition 4.1 of [22].
Proof of Lemma 4.3. Under the assumptions of Lemma 4.3, by Bishop–Gromov compari-
son theorem, we have vol(M) ≤ V = V (n,Λ, D). Taking δ1 = δ2(ε, n,Λ, i0, V (n,Λ, D)), the
condition of Lemma 4.4 is satisfied.
Since ‖e−v‖∞ ≤ E, we have
∫
M |G|
pdm =
∫
M |G|
pe−vdvol ≤ ECp(n,Λ, D, i0)
p. Hence
‖G‖p ≤ C(p, n,Λ, E,D, i0).
Since Ric∞,m = Ric +∇2v, we have∫ dφ
0
Ric(γ̇) dt =
∫ dφ
0
Ric∞,m(γ̇) dt−
∫ dφ
0
∇2v(γ̇, γ̇) dt
=
∫ dφ
0
Ric∞,m(γ̇) dt−
∫ dφ
0
〈∇γ̇∇v, γ̇〉 dt
=
∫ dφ
0
Ric∞,m(γ̇) dt−
∫ dφ
0
(
d
dt
〈∇v, γ̇〉 − 〈∇v,∇γ̇ γ̇〉
)
dt
=
∫ dφ
0
Ric∞,m(γ̇) dt− 〈∇v(γ(dφ)), γ̇(dφ)〉+ 〈∇v(x), γ̇(0)〉. (4.6)
On the other hand, for any x 6= φ(x) we are going to calculate ∇dφ(x). For the sake of clarity, we
denote rp(x) = d(p, x) where d is the metric on M , and x, p ∈ M are two points. Then for any
x 6= φ(x), ∇dφ(x) = ∇rφ(x)(x) +∇(rx ◦ φ)(x) = −γ̇(0) + (φ∗)
tγ̇(dφ) where φ∗ : TxM → Tφ(x)M
is the tangent map and (φ∗)
t : Tφ(x)M → TxM is the transpose of the tangent map defined by
〈(φ∗)tU,W 〉 = 〈U, (φ∗)W 〉 for all U ∈ Tφ(x)M and W ∈ TxM . Therefore, we have
〈∇v(x),∇dφ(x)〉 =
〈
∇v(x),−γ̇(0) + (φ∗)
tγ̇(dφ)
〉
= 〈(φ∗∇v)(γ(dφ)), γ̇(dφ)〉 − 〈∇v(x), γ̇(0)〉
= 〈∇v(γ(dφ)), γ̇(dφ)〉 − 〈∇v(x), γ̇(0)〉 , (4.7)
where in the last equality we use the fact that φ is a measure preserving isometry. Plugging (4.6)
and (4.7) into (4.5), we get for all x 6= φ(x)
∆mdφ(x) ≥ −
∫ dφ(x)
0
Ric∞,m(γ̇) dt− εG(x)dφ(x) ≥ −dφ(x)ρφ(x)− εG(x)dφ(x).
From this pointwise inequality, with the same argument as the proof at the very end of [10], we
have in the sense of distribution ∆mdφ ≥ −dφρφ − εGdφ. �
The Measure Preserving Isometry Groups of Metric Measure Spaces 13
Proof of Theorem 1.7. Since our Lemma 4.3 has exactly the same form as [22, Proposi-
tion 4.1] with the only difference in the Laplacian, changing the volume element in every integral
and Lp norm in the proof of Theorem 0.3 in [22] to the measure m, we get the conclusion.
For tha sake of completeness, we write down the proof here. Similar to Theorem 1.6, it suffices
to prove that under the conditions of Theorem 1.7 there exists δ = δ(n, i0,Λ1,Λ2, E,D,w,A,B)
> 0, such that for any measure preserving isometry φ with δφ ≤ δ, we have φ = id. Then the
remaining part of the proof is the same as the proof of Theorem 1.6.
We assume δφ ≤ δ ≤ δ1 so that Lemma 4.3 is valid. Multiplying dφ to (4.4) and integrating
over M with respect to the measure m, we have
0 ≥ −
∫
M
(∆mdφ + ρφdφ + εGdφ)dφ dm
≥ ‖Ddφ‖22 −
∫
M
[(ρφ + w)+ + εG]d2φ dm+ w ‖dφ‖22
≥ ‖Ddφ‖22 − (‖(ρφ + w)+‖n/2 + ε‖G‖n/2)‖dφ‖22n/(n−2) + w‖dφ‖22
≥ ‖Ddφ‖22 −
(
Λ2V
2/nδφ + ‖(θ∗ + w)+‖n/2 + ε‖G‖n/2
)
‖dφ‖22n/(n−2) + w‖dφ‖22 (4.8)
≥
[
1−A
(
Λ2V
2/nδφ + ‖(θ∗ + w)+‖n/2 + ε‖G‖n/2
)]
‖Ddφ‖22
+
[
w −B
(
Λ2V
2/nδφ + ‖(θ∗ + w)+‖n/2 + ε‖G‖n/2
)]
‖dφ‖22, (4.9)
where in (4.8) we have used (4.3) and in (4.9) we have used the Sobolev inequality (1.3). If we
take
δφ ≤ δ = min
{
δ1,
1
4AΛ2V 2/n
,
w
4BΛ2V 2/n
}
, ε =
C (n/2, n,Λ1, E,D, i0)
4
min
{
1
A
,
w
B
}
then we have
0 ≥
(
1
2
−A ‖(θ∗ + w)+‖n/2
)
‖Ddφ‖22 +
(w
2
−B ‖(θ∗ + w)+‖n/2
)
‖dφ‖22 .
From the assumption of w, the coefficients of ‖Ddφ‖22, ‖dφ‖22 are positive, so we have dφ ≡ 0,
i.e., φ = id. �
Acknowledgements
The results in this article are mainly part of the author’s undergraduate thesis at Tsinghua
University. The author would like to express his sincere gratitude to Professor Jinxin Xue who
brought him into this field and gave him expert advice. He would also like to thank Professors
Yann Brenier and Francois Bolley for their email of discussion and Professor Tapio Rajala for
telling him the articles [17, 28] on the measure preserving isometry groups of RCD spaces.
Finally, he would like to thank the anonymous referees for their useful comments which leads to
Theorem 1.7.
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1 Introduction
1.1 Synthetic Ricci curvature bounds
1.2 Bochner's theorem
2 Synthetic Ricci curvature upper bounds
3 Generalization of Bochner's theorem to metric measure spaces
4 Estimating the order of the measure preserving isometry group
References
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| id | nasplib_isofts_kiev_ua-123456789-211006 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T09:12:28Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Guo, Yifan 2025-12-22T09:24:36Z 2020 The Measure Preserving Isometry Groups of Metric Measure Spaces. Yifan Guo. SIGMA 16 (2020), 114, 14 pages 1815-0659 2020 Mathematics Subject Classification: 53C20; 53C21; 53C23 arXiv:2006.04092 https://nasplib.isofts.kiev.ua/handle/123456789/211006 https://doi.org/10.3842/SIGMA.2020.114 Bochner's theorem says that if is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso( ) is finite. In this article, we show that if ( , , ) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure-preserving isometry group Iso( , , ) is finite. We also give an effective estimate on the order of the measure-preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Émery Ricci curvature, except for small portions. The results in this article are mainly part of the author's undergraduate thesis at Tsinghua University. The author would like to express his sincere gratitude to Professor Jinxin Xue, who brought him into this field and gave him expert advice. He would also like to thank Professors Yann Brenier and Francois Bolley for their email of discussion and Professor Tapio Rajala for telling him the articles [17, 28] on the measure-preserving isometry groups of RCD spaces. Finally, he would like to thank the anonymous referees for their useful comments, which led to Theorem 1.7. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Measure Preserving Isometry Groups of Metric Measure Spaces Article published earlier |
| spellingShingle | The Measure Preserving Isometry Groups of Metric Measure Spaces Guo, Yifan |
| title | The Measure Preserving Isometry Groups of Metric Measure Spaces |
| title_full | The Measure Preserving Isometry Groups of Metric Measure Spaces |
| title_fullStr | The Measure Preserving Isometry Groups of Metric Measure Spaces |
| title_full_unstemmed | The Measure Preserving Isometry Groups of Metric Measure Spaces |
| title_short | The Measure Preserving Isometry Groups of Metric Measure Spaces |
| title_sort | measure preserving isometry groups of metric measure spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211006 |
| work_keys_str_mv | AT guoyifan themeasurepreservingisometrygroupsofmetricmeasurespaces AT guoyifan measurepreservingisometrygroupsofmetricmeasurespaces |