The measure preserving isometry groups of metric measure spaces

Bochner's theorem says that if $M$ is a compact Riemannian manifold with negative Ricci curvature then the isometry group $\mathrm{Iso}(M)$ 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 $\mathrm{Iso}(X,d,m)$ is finite. We also give effective estimate on the order of measure preserving isometry group for compact weighted Riemannian manifold with an integral bound on Bakry-Emery Ricci curvature.


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 weighted Riemannian manifold (M, g, m), which means that (M, g) is a Riemannian manifold, and m = e −V vol g where vol g is the volume element associated with g and V : M → R is a C 3 function. In this case, X = M , d = d g is the intrinsic metric induced by g and m = e −V vol g . For N ∈ [n, +∞) we define the N -Bakry-Emery Ricci tensor on (M, g, m) by Ric N,m := Ric + ∇ 2 V − dV ⊗ dV N − n with the convention that when N = n, we require V = 0 so that Ric n,m = Ric. In case of N = +∞, we define Ric ∞,m := Ric + ∇ 2 V. Also, we define the Witten Laplacian ∆ m u := ∆ g u − ∇V · ∇u.
It is natural to ask whether there is a synthetic notion of Ricci curvature upper bound. This question is addressed by Sturm in his paper [32]. Sturm's definition is inspired by an equivalent definition of an infinitesimally Hilbertian space (X, d, m) satisfying CD(K, ∞) condition (c.f. [3]): the inequality holds for all x, y ∈ M and t > 0 . Here, W is the 2-Wasserstein distance on P 2 (X) and H t is the heat flow or gradient flow of entropy functional Ent m , while δ x is the Dirac measure at x (see section 2 for definitions). The basic idea is to replace " ≤ " by " ≥ " in (1). The following quantities are important From (1), it can be shown that RCD(K, ∞) is equivalent to θ + (x, y) ≥ K for all x, y ∈ X (c.f. [32,Theorem 2.10]). If (M, g, m) is a weighted Riemannian manifold, Sturm [32] proved that θ * (x) = sup{Ric ∞,m (ξ, ξ)/|ξ| 2 : ξ ∈ T x M } for all x ∈ M . This motivates the following definition. (See section 2 for more details.) Definition 1.1 (Sturm [32]). We say that the metric measure space (X, d, m) has synthetic However, it should be noted that due to the theorem of Gao-Yau [13] and Lohkamp [24], Ricci curvature upper bound does not place any topological restriction on the manifold. More precisely, for any integer n ≥ 3, there exists two positive constants Λ 1 = Λ 1 (n), Λ 2 = Λ 2 (n) such that each manifold M of dimension n admits a complete metric g with −Λ 1 < Ric(g) < −Λ 2 .
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 bound as Ricci curvature lower bound. In fact, it was pointed out by Gromov (see [17, §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 compact manifold with negative Ricci curvature (Ricci curvature bounded above by 0) is Bochner's theorem (c.f. [10]) which states that the isometry group of the manifold must be finite. In fact, Bochner showed that there exists no continuous group of isometry by considering the Laplacian of square norm of Killing vector field. This technique is the genesis of the famous "Bochner technique" which produces numerous geometric 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 [24]). 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 [9] showed that when X is a Riemann surface of genus g ≥ 2, | Aut(X)| ≤ 84(g − 1). Later, the estimate of the order of isometry group was generalized to hyperbolic manifolds by Huber [20], to manifolds with sectional curvature bounded above from 0 by Im Hof [21], to manifolds with non-positive sectional curvature and Ricci curvature negative at some point by Maeda [26] and to manifolds with non-positive sectional curvature and finite volume by Yamaguchi [35]. For general compact Riemannian manifolds with negative Ricci curvature, Katsuda [22] estimated the order of isometry group by sectional curvature, dimension, diameter and injectivity radius. In [12], Dai-Shen-Wei estimated the order by Ricci bounds, volume, injectivity radius. Recently, Katsuda-Kobayashi [23] gave a bound of isometry group by computable constant assuming only integral bound on Ricci curvature.
There are also other generalizations of the Bochner's theorem. In [28], Rong showed that compact manifolds with negative Ricci curvature does not admit non-trivial invariant F -structure which includes Killing vector field as a special case. Bagaev and Zhukova [7] extended Bochner's theorem to Riemannian orbifolds with negative Ricci curvature. From the works of Deng-Hou [11] and Zhong-Zhong [37], we know that a compact Finsler manifold with negative Ricci curvature has a finite isometry group. Van Limbeek [31] estimated the order of isometry group for manifolds on which a circle action does 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.
As a corollary, we have the Bochner theorem for weighted Riemannian manifolds: This corollary can also be obtained by considering the Laplacian of the 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 weighted Riemannian manifold of dimension n. If the N -Bakry-Emery Ricci tensor Ric N,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 the Bochner theorem should hold on spaces with synthetic negative N -Ricci curvature.
Our next result generalizes the theorems in Dai-Shen-Wei [12] and Katsuda-Kobayashi [23] to weighted Riemannian manifold with integral bound on ∞-Bakry-Emery Ricci curvature. We denote the sectional curvature by K M , the injectivity radius by inj M , and the L p (M, m) norm by · p i.e. for any Borel function f : for some fixed positive constants i 0 , N, where (θ * + w) + = max(θ * + w, 0), and A, B are constants such that the Sobolev inequality holds for all f ∈ W 1,2 (M, g, m), then there exists a constant L = L(n, i 0 , N, It would be interesting to know whether there is an estimate of the order of measure preserving isometry group on metric measure space instead of weighted Riemannian manifold. Our proof of theorem 1.6 is a first step in this direction. We use optimal transport to prove a key lemma (lemma 4.1) which is an analogue of Lemma 4.2 in [12] and Proposition 2.2 in [23]. However, the difficulty of going further is to relate θ + (x, y) and θ * (x) using geometric quantities of the metric measure space.
The plan of the rest of the paper is as follows. In section 2, we introduce synthetic Ricci curvature upper bound. In section 3, we prove our generalization of Bochner theorem. In section 4, we give our estimate on the order of the measure preserving isometry group.

Ricci curvature upper bound
Let (X, d, m) be a metric measure space. We denote by P (X) the set of all Borel probability measures on X. We define the 2-Wasserstein distance on P 2 (X) = {µ ∈ P (X) : where the infimum is taken among all π ∈ P (X × X) such that π has marginals µ and ν. Such π is called an admissable plan. The measure π at which the infimum is realized is called an optimal transport plan of µ and ν, which always exists. (see [30,Chapter 4]).
We define the entropy functional on P (X) by Let K ∈ R, N > 1 be two numbers. For t ∈ [0, 1], we define the functions β Definition 2.1. We say that a metric measure space X satisfies curvature dimension condition CD(K, ∞) if for any two measures µ 0 and µ 1 in P 2 (X), there exists some geodesic A metric measure space (X, d, m) is said to satisfy curvature dimension condition CD(K, N ) if for any two measures µ 0 and µ 1 in P 2 (X), there exists some geodesic (µ t = ρ t m) in P 2 (X) such that for some optimal transport plan π of µ 0 and µ 1 and all 0 ≤ t ≤ 1 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, Ric N,m ≥ K (see Lott-Villani [25], Sturm [33], [34]).
Let (X, d, m) be a metric measure space and f ∈ L 2 (X, m). We define the Cheeger energy by Being a convex lower semicontinuous function, the Cheeger energy admits a gradient flow H t on L 2 (X, m). However, the Cheeger energy Ch is not neccessarily a quadratic form. We say that (X, d, m) is infinitesimal Hilbertian if Ch is a quadratic form, i.e.
We also denote by H t the metric gradient flow of Ent m on P (X). For more details on metric gradient flow, see e.g. [4]. [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 L 2 (X, m).

Definition 2.2 (Ambrosio-Gigli-Savaré [3], Ambrosio-Gigli-Modino-Rajala [1], Gigli
In RCD(K, ∞) spaces, heat flow behaves well. It is the result of Ambrosio-Gigli-Savaré [2,3] that the gradient flow H t of Ch and the gradient flow H t of Ent m coincides and that for every f ∈ L 2 (X, m) ∩ L ∞ (X, m), and every x ∈ X we have Moreover, the heat flow is a EVI K gradient flow i.e. the heat flow satisfies the contraction property (1). Turning to the Ricci curvature upper bound, we recall the definition of For a function u : R → R, we write ∂ − t u := lim inf s→t (u(s) − u(t))/(s − t). Thus θ + (x, y) = − ∂ − t t=0 log W (H t δ x , H t δ y ). 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 Sturm has proved the following two sided bound for θ + (x, y). Theorem 2.3 (Sturm [32]). Let (M, g, m) be a 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 where d is the metric induced by g, and for R the Riemann curvature tensor and {e i } n i=1 some parallel orthonormal frame along γ such that e 1 =γ.

Taking the endpoints infinitely close to each other in (3), we have
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

Generalization of Bochner's theorem to metric measure space
In this section we will prove our generalization of Bochner's theorem. A fundamental lemma is the following discrete version of Bochner's theorem which is a standard technique (c.f. [22]). 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.
Let y ∈ X and z = φ(y) be such that d(y, z) = max x∈X d(x, φ(x)) ≤ λ. There exists Hence for t > 0 small, On the other hand, since φ is an isometry that preserves the measure m, φ # (H t δ y ) = H t (φ # δ y ) = H t δ φ(y) = H t δ z . Therefore, (id×φ) # (H t δ y ) is an admissible plan of (H t δ y , H t δ z ). Hence for all t > 0 Therefore, we get, for t > 0 small, d 2 (y, z) ≥ e ǫt d 2 (y, z), which is impossible unless d(y, z) = 0 i.e. φ is the identity map. Now, we can prove our main theorem.
Proof of Theorem 1.3. Take λ as in lemma 3.1 and a = λ/4. Let x 1 , . . . , x N ∈ X be such that We define a map F from the measure preserving isometry group By the above lemma, φ = ψ. Hence F is injective and we get 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 measure preserving isometry group Iso(M, g, m) is a compact Lie group. (In general, the measure preserving isometry groups of RCD * (K, N ) spaces is Lie group. See [19,29].) 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 φ t of ξ preserves g and m, i.e. (φ t ) * g = g, (φ t ) # m = m. Hence we have for all vector fields U, W on M ∇ U ξ, W + ∇ W ξ, U = 0, div m ξ = div g ξ − ∇V, ξ = 0, where , denotes g and div g is the divergence operator associated with g. Since ∇ξ is antisymmetric, div g ξ = tr(∇ξ) = 0, which implies ∇V, ξ = 0. Hence we obtain Hence we get 1 2 ∆ m |ξ| 2 = |∇ξ| 2 − Ric ∞,m (ξ, ξ).

Estimating the order of measure preserving isometry group
In this section, we give an explicit dependence of the upper bound in (5). 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 d φ (x) = d(x, φ(x)) and δ φ = 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 δ φ ≤ inj M ), then we define Proof. Let φ : M → M be a measure preserving isometry. With the same argument as (4), we have where we have used the formula (2) in the last equality.
Since lim and hence Since M is compact, it automatically satisfies some RCD(−K ′ , N ) condition. Choosing δ ≤ i 0 and applying the upper bound estimate in (3) to x, φ(x), we have Since |K M | ≤ Λ, we have |R(e i ,γ, e j ,γ)| ≤ 2Λ so that Combining (7), we get The following lemma is a quantitative version of Lemma 3.1. If for some w > 0 (θ * + w) + n/2 < min 1 4A , w 4B .