Cheng maximal diameter theorem for hypergraphs
CCHENG’S MAXIMAL DIAMETER THEOREM FORHYPERGRAPHS
YU KITABEPPU AND ERINA MATSUMOTO
Abstract.
We prove that Cheng’s maximal diameter theorem for hyper-graphs with positive coarse Ricci curvature. Introduction
It holds that the diameter of an n -dimensional Riemannian manifold with Riccicurvature bounded below by K ( n − > π/ √ K by the famous Bonnet-Myers theorem. Moreover, under the same assumption and if Diam M = π/ √ K ,then M is isometric to the n dimensional sphere S n ( K − / ), which is known asCheng’s maximal diameter theorem. These theorem have been extended to manyother situations(see for instance [7–10, 15, 20–23, 28, 30, 31, 36] for Bonnet-Myerstheorem and [12, 17, 19, 29, 32, 33] for rigidity theorem). Here we focus on Bonnet-Myers type and Cheng’s maximal diameter theorems for CD spaces and RCD spaces.A synthetic notion of Ricci curvature bounded from below and dimension boundedfrom above on generic metric measure spaces, called the Curvature-Dimension con-dition CD ( K, N ) ( K ∈ R , N ∈ [1 , ∞ ]), was introduced by Sturm and Lott-Villani[23,35,36]. Though many geometric and analytic properties are proven such spaces,the class of CD spaces is seen to be still huge. In fact, Finsler manifolds can be-come CD spaces and the Laplacian and the heat flow with respect to the Cheegerenergy on generic CD spaces are possibly nonlinear. To restrict CD spaces to bemore a manageable class, RCD spaces are defined by satisfying CD condition withthe infinitesimally Hilbertian condition [3–5, 13]. The heat flow and the Laplacianbecome linear operators thanks to the infinitesimal Hilbertianity. It is proven thaton infinitesimal Hilbertian space, CD condition and Bakry-´Emery type curvature-dimension condition coincides under mild assumptions [1]. For finite dimensional CD spaces, Bonnet-Myers theorem holds [23, 36]. On the other hand, the maximaldiameter theorem have not been proven on CD spaces(under the non-branching as-sumption, CD spaces are MCP ones. Therefore a partial result is given in [29]. Onthe other hand,
MCP condition is not enough to guarantee the space being sphericalsuspension. see [18]). However on
RCD spaces, Cheng’s maximal diameter theoremholds [17]. In [17], the infinitesimal Hilbertianity(or the linearity of the Laplacian)seems to play an important role in the proof.There exist many notions of Ricci curvature on graphs(for example, [6, 10, 11,14, 22, 24, 26, 27, 30, 31, 34]). The positivity of lower bound of Ricci curvature oftenleads the bounds of diameter. And sometimes Cheng’s maximal diameter theoremis also given. For instance, the maximal diameter theorem on graphs and directedgraphs under positive Ricci curvature in the sense of Lin-Lu-Yau are proven in[12, 32]. Note that Laplacian related such notions are linear though sometimesthose operator is not symmetric and the corresponding Dirichlet form is nonlocal.Hence can we guess the linearity of the Laplacian needs to guarantee holding themaximal diameter theorem?
Key words and phrases. hypergraph, coarse Ricci curvature. a r X i v : . [ m a t h . M G ] F e b YU KITABEPPU AND ERINA MATSUMOTO
In the present paper, we consider Cheng’s maximal diameter theorem underpositive bound of coarse Ricci curvature on hypergraphs. The first author, Ikeda,and Takai introduced a Lin-Lu-Yau’s type coarse Ricci curvature notion on theclass of weighted hypergraphs [15]. Under positive Ricci curvature bound, we haveBonnet-Myers type diameter bound. Actually if inf x (cid:54) = y κ ( x, y ) ≥ K > Diam V ≤ /K holds. Although our definition of Ricci curvature is based on the resolventoperator associated with the Laplacian that is nonlinear in general, we have thefollowing theorem. Theorem 1.1.
Let H := ( V, E, ω ) be a finite weighted hypergraph. Assume that inf x (cid:54) = y κ ( x, y ) ≥ K > . Accordingly Diam V ≤ /K . Moreover assume Diam V =2 /K . Then there exists a pair of points p, q ∈ V with d ( p, q ) = Diam V such thatevery point v ∈ V is on a geodesic path connecting p and q . Moreover if x, y ∈ V are on the same geodesic path connecting p to q , then κ ( x, y ) = κ ( x, y ) = K . Preliminaries
Monotone operator.
In this subsection, we give some results of the theoryof maximal monotone operators that we need later. See [25] for the detailed proofsor more reference.Let H be a Hilbert space with the inner product (cid:104)· , ·(cid:105) and A a multivaluedoperator, that is, Ax ⊂ H for any x in the domain of A , denoted by D ( A ). Wedenote the range of A by R ( A ) := { y ; y ∈ Ax for some x ∈ D ( A ) } . And we definethe inverse of A by A − x = { y ; x ∈ Ay } with D ( A − ) = R ( A ). We say that A is monotone if (cid:104) x (cid:48) − y (cid:48) , x − y (cid:105) ≥ x, y ∈ D ( A ) and any x (cid:48) ∈ Ax , y (cid:48) ∈ Ay . A monotone operator A iscalled an m -monotone operator if R ( I + λA ) = H for any λ >
0, where I is theidentity operator on H . It is known that A is an m -monotone operator if and only if A is a maximal monotone one. Here a monotone operator A is said to be maximal ifany monotone operator B satisfying D ( A ) ⊂ D ( B ) and Ax ⊂ Bx for any x ∈ D ( A )coincides A .From now on, we always assume that A is an m -monotone operator. It is knownthat Ax is a closed convex subset in H for any x ∈ D ( A ). Hence we find a uniqueelement ˜ x = argmin {(cid:107) x (cid:48) (cid:107) ; x (cid:48) ∈ Ax } . We call ˜ x the canonical restriction of A anddenote it by A x . Since R ( I + λA ) = H , we are able to define the resolvent operators J λ := ( I + λA ) − . One notable property of J λ is that is actually single-valued. Wealso know the following result. Theorem 2.1.
Let A be an m -monotone operator. Then we have a unique solution u ( t ) to ddt u ( t ) ∈ − Au ( t ) , a.e. t > u (0) = x ∈ D ( A ) . Moreover it is equivalent to the existence of a unique solution to d + dt u ( t ) = − A u ( t ) , t ≥ u (0) = x ∈ D ( A ) . These two solutions are absolutely continuous and coincide with each other.
By the result above, we have the contraction semigroup with the generator A ,that is, there exists a family of single-valued operators { T t } t> with D ( T t ) = D ( A )such that • T = I , T s + t = T s T t , • For x ∈ D ( T t ), T t x is strongly continuous with respect to t , HENG’S MAXIMAL DIAMETER THEOREM FOR HYPERGRAPHS 3 • (cid:107) T t x − T t y (cid:107) ≤ (cid:107) x − y (cid:107) , • lim h ↓ h ( T h x − x ) = − A x holds for any x ∈ D ( A ).In general T t is not a linear operator. Example 2.2.
Let E : H → R ∪ {∞} be a convex and lower semi-continuousfunctional on a real Hilbert space ( H, (cid:104)· , ·(cid:105) ). Set D ( E ) := { f ∈ H ; E ( f ) < ∞} . h ∈ H is a sub-differential of E at f provided that E ( g ) − E ( f ) ≥ (cid:104) h, g − f (cid:105) holds for any g ∈ H . The set of all sub-differentials of E at f is denoted by ∂E ( f ). Then it is known that ∂E : H → H is an m -monotone operator defined on D ( ∂E ) := { f ∈ D ( E ) ; ∂E ( f ) (cid:54) = ∅} .For this case, we have another representation of J λ . It actually satisfies J λ f = argmin (cid:26) (cid:107) f − g (cid:107) λ + E ( g ) ; g ∈ D ( E ) (cid:27) . Hypergraph.
A hypergraph H = ( V, E ) consists of a vertex set V and ahyperedge set E ⊂ V . Unlike in the case of a graph, e ∈ E can include more thantwo vertices. We use the notation x ∼ y for x, y ∈ V provided x, y is included ina same hyperedge e ∈ E . For given two vertices x, y ∈ V , we define the distancebetween them by d ( x, y ) := min { n ; x = x , x i ∼ x i +1 , x n = y for i = 1 , · · · , n − } . A sequence of vertices { x i } ni =0 that realizes the distance d ( x , x n ) is called a geodesicpath . When | V | < ∞ , H = ( V, E ) is called a finite hypergraph. A weight function ω : E → R > is denoted by ω e or ω ( e ) for e ∈ E . The weighted degree d x at x ∈ V is defined by d x := (cid:80) e (cid:51) x ω e . Set a matrix D = ( D u,v ) u,v ∈ V ∈ R V × V by D u,v := (cid:40) d u if u = v, , and call it the (weighted) degree matrix . In this article, we consider only finiteweighted hypergraphs.Since | V | < ∞ , a function f : V → R is identified with a column vector f ∈ R V .An inner product for f, g ∈ R V is defined by (cid:104) f, g (cid:105) := (cid:88) x ∈ V f ( x ) g ( x ) d − x . We define a function δ v ∈ R V for v ∈ V by δ v ( u ) = (cid:40) u = v, . Given f ∈ R V , the base polytope B e for e ∈ E is defined as B e = Conv { δ u − δ v ; u, v ∈ e } ⊂ R V . We say a function f : V → R weighted K -Lipschitz for K > (cid:104) f, δ x − δ y (cid:105) ≤ Kd ( x, y ) holds for any x, y ∈ V and denote the set of weighted K -Lipschitz functionsby Lip Kω ( V ). YU KITABEPPU AND ERINA MATSUMOTO
We define the
Laplacian on the functions defined on V as a multivalued operatorby L f := (cid:40)(cid:88) e ∈ E ω e b T e ( D − f ) b e ; b e ∈ argmax b ∈ B e b T ( D − f ) (cid:41) , here b T is the transpose vector of b . The following theorem is known. Theorem 2.3. [16, 37]
The Laplacian L is an m -monotone operator on the Hilbertspace ( R V , (cid:104)· , ·(cid:105) ) . Hence we define the resolvent operator J λ : R V → R V and the contractionsemigroup h t : R V → R V , that are both single-valued. We call { h t } t> the heatsemigroup or heat flow . One holds the following property, that is, L ( af ) = a L f, J λ ( af ) = aJ λ f, h t ( af ) = ah t ( f )(2.1)for f ∈ R V , a ∈ R . Remark . Actually L is defined as the sub-differential of a functional on R V .Let n = | V | , N := | E | . Define a functional E ( f ) := 12 (cid:88) e ∈ E ω e (cid:18) max u ∈ e D − f ( u ) − min v ∈ e D − f ( v ) (cid:19) . For any f (cid:48) = (cid:80) e ∈ E ω e b T e ( D − f ) b e ∈ L f , b e ∈ argmax b ∈ B e b T ( D − f ), we denote itby f (cid:48) = BW B T ˜ f , where B = [ b e , · · · , b e N ] ∈ R n × N , W = diag( ω e , · · · , ω e N ) ∈ R N × N , ˜ f = D − f ∈ R V = R n . E ( f ) = 12 (cid:88) e ∈ E ω e (cid:0) b T e ( D − f ) (cid:1) = 12 ( ˜ f ) T BW B T ˜ f = 12 (cid:68) BW B T ˜ f , f (cid:69) = 12 (cid:104) f (cid:48) , f (cid:105) holds. The continuity and the convexity of E are clear. For k, h ∈ R E , we define (cid:104) h, k (cid:105) W := h T W k . It is an inner product that satisfies (cid:104) f (cid:48) , f (cid:105) = (cid:68) B T ˜ f , B T ˜ f (cid:69) W .Then for any g ∈ R V , g (cid:48) = B W B T ˜ g ∈ L g , it holds (cid:104) f (cid:48) , g − f (cid:105) = (cid:104) f (cid:48) , g (cid:105) − (cid:104) f (cid:48) , f (cid:105) = (cid:68) B T ˜ f , B T ˜ g (cid:69) W − (cid:104) f (cid:48) , f (cid:105)≤ (cid:107) B T ˜ f (cid:107) W (cid:107) B T ˜ g (cid:107) W − (cid:104) f (cid:48) , f (cid:105)≤ (cid:107) B T ˜ f (cid:107) W + 12 (cid:107) B T ˜ g (cid:107) W − (cid:104) f (cid:48) , f (cid:105)≤ (cid:107) B T ˜ g (cid:107) W − (cid:107) B T ˜ f (cid:107) W = E ( g ) − E ( f ) . This means f (cid:48) ∈ ∂E ( f ). Accordingly L f ⊂ ∂E ( f ). Since both L and ∂E aremaximal monotone operators, L = ∂E .1l ∈ R V is defined by 1l( x ) = 1 for any x ∈ V . Lemma 2.5.
For any f ∈ R V and any f (cid:48) ∈ L f , (cid:104) f (cid:48) , D (cid:105) = 0 . Proof.
By the definition of the sub-differentials, we have0 = E ( f + D − E ( f ) ≥ (cid:104) f (cid:48) , D (cid:105) , E ( f − D − E ( f ) ≥ (cid:104) f (cid:48) , − D (cid:105) . Combining these inequalities leads the consequence. (cid:3)
HENG’S MAXIMAL DIAMETER THEOREM FOR HYPERGRAPHS 5
Coarse Ricci curvature on hypergraphs.
Let H = ( V, E, ω ) be a weightedfinite hypergraph. We define the new metric KD λ ( x, y ) on V , which plays a role ofthe L -Wasserstein distance between random walks with the initial distributions δ x and δ y respectively. Definition 2.6 (Nonlinear Kantorovich difference [15]) . Given λ > x, y ∈ V .The λ -nonlinear Kantorovich difference KD λ ( x, y ) is defined as KD λ ( x, y ) := sup (cid:8) (cid:104) J λ f, δ x − δ y (cid:105) ; f ∈ Lip ω ( V ) (cid:9) . Remark . Each KD λ is a metric on V ([15]). They are defined to imitate theKantorovich-Rubinstein duality formula. We take a devious route not to tackle aproblem to determine what is the canonical random walk on a hypergraph. Rela-tions between KD λ and KD µ for λ, µ > Definition 2.8 (Coarse Ricci curvature [15]) . λ -coarse Ricci curvature κ λ ( x, y )( x, y ∈ V ) is defined by κ λ ( x, y ) := 1 − KD λ ( x, y ) d ( x, y ) . The upper coarse Ricci curvature κ is defined by κ ( x, y ) := lim sup λ → +0 κ λ ( x, y ) λ . Similarly, the lower coarse Ricci curvature κ is defined by κ ( x, y ) := lim inf λ → +0 κ λ ( x, y ) λ . Remark . For a positive constant a >
0, hypergraphs H = ( V, E, ω ) and H (cid:48) =( V, E, aω ) have the same curvature. Indeed, (cid:104)· , ·(cid:105) (cid:48) , J (cid:48) λ , KD (cid:48) λ and Lip aω ( V ) is theinner product, the resolvent operator, λ -nonlinear Kantorovich difference and theset of weighted 1-Lipschitz functions on H (cid:48) respectively. It is easy to verify J λ f = J (cid:48) λ f and f ∈ Lip ω ( V ) iff af ∈ Lip aω ( V ). Therefore, for f (cid:48) = af ∈ Lip aω ( V ) with f ∈ Lip ω ( V ), we have (cid:104) J (cid:48) λ ( f (cid:48) ) , δ x − δ y (cid:105) (cid:48) = a − (cid:104) J λ ( af ) , δ x − δ y (cid:105) = (cid:104) J λ f, δ x − δ y (cid:105) . Hence KD (cid:48) λ ( x, y ) = KD λ ( x, y ). Accordingly λ -coarse Ricci curvature coincides.We have the following formula. Lemma 2.10.
Let H = ( V, E, ω ) be a weighted hypergraph. Then κ ( x, y ) ≤ d ( x, y ) C ( x, y ) , (2.2) where C ( x, y ) = inf (cid:8)(cid:10) L f, δ x − δ y (cid:11) ; f ∈ Lip ω ( V ) , (cid:104) f, δ x − δ y (cid:105) = 1 (cid:9) . Lemma 2.11.
Let f ∈ Lip w ( V ) be a weighted 1-Lipschitz function and f (cid:48) ∈ L f .Then | f (cid:48) ( x ) | ≤ d x holds for any x ∈ V . Theorem 2.12 (Bonnet-Myers type theorem) . Let H = ( V, E, ω ) be a weightedhypergraph with positive upper coarse Ricci curvature bound inf u (cid:54) = v κ ( u, v ) ≥ K > .Then Diam H ≤ /K . YU KITABEPPU AND ERINA MATSUMOTO
The proof is already given in [15]. Since some techniques used in the proof areuseful even for the main theorem, we show the outline of the proof.
Proof.
Let p, q ∈ V be points such that d ( p, q ) = Diam H . For any f ∈ Lip ω ( V ), |L f ( x ) | ≤ d x by Lemma 2.11. Then if (cid:104) f, δ p − δ q (cid:105) = d ( p, q ), K ≤ κ ( p, q ) ≤ (cid:10) L f, δ p − δ q (cid:11) d ( p, q ) ≤ (cid:10) |L f | , δ p + δ q (cid:11) Diam H ≤ Diam
H . (cid:3)
Remark . Since κ ( u, v ) ≤ κ ( u, v ) holds for any u, v ∈ V , the assumption inTheorem 2.12 can be replaced by inf u (cid:54) = v κ ( u, v ) ≥ K instead of the lower bound ofupper coarse Ricci curvature.3. Proof of main theorem
We prove the following theorem.
Theorem 3.1.
Let H = ( V, E, ω ) be a weighted hypergraph with positive lowercoarse Ricci curvature bound inf u (cid:54) = v κ ( u, v ) ≥ K > . Assume Diam H = 2 /K .Then there exists a pair of points p, q ∈ V such that every point x ∈ V is on ageodesic path from p to q . Moreover if x, y ∈ V are on the same geodesic path.Then κ ( x, y ) = κ ( x, y ) ≡ K .Proof. Let p, q ∈ V be points such that d ( p, q ) = Diam H = 2 /K =: L . Definefunctions ρ p , ρ q , R : V → R by ρ p ( x ) := d x r p ( x ) := d x d ( p, x ) , ρ q ( x ) := d x d ( x, q ) , R ( x ) := Ld x . Put L ρ p = (cid:80) e ω e b e b Te ( r p ). By the proof of Theorem 2.12, we have | (cid:10) L ρ p , δ q − δ p (cid:11) | =2. Since ρ p ( p ) = min x ρ p ( x ), we have L ρ p ( p ) = (cid:80) e (cid:51) p ω e ( −
1) = − d p . Combiningthese two facts, we obtain2 = (cid:10) L ρ p , δ q − δ p (cid:11) = (cid:10) L ρ p , δ q (cid:11) + 1 ⇔ (cid:10) L ρ p , δ q (cid:11) = 1 . By the same way, we also have (cid:10) L ρ q , δ p (cid:11) = − (cid:10) L ρ q , δ q (cid:11) = 1.Suppose there exists a point v ∈ V such that f ( v ) := ρ p ( v ) + ρ q ( v ) − R ( v ) ≥ d v > . (3.1)By (2.2), K ≤ κ ( p, v ) ≤ (cid:10) L ρ p , δ v − δ p (cid:11) d ( p, v ) ⇔ Kr p ( v ) ≤ (cid:10) L ρ p , δ v (cid:11) + 1 . Multiplying d v leads Kρ p ( v ) ≤ L ρ p ( v ) + d v . (3.2)By a similar calculation, we have Kρ q ( v ) ≤ L ρ q ( v ) + d v . (3.3)Combining (3.1), (3.2), and (3.3), we obtain L ρ p ( v ) + L ρ q ( v ) + 2 d v ≥ K ( ρ p ( v ) + ρ q ( v )) = K ( R ( v ) + f ( v ))= 2 d v + 2 L f ( v ) . Set A := { v ; f ( v ) > } . Summing up v ∈ V leads, (cid:88) v ∈ V (cid:0) L ρ p ( v ) + L ρ q ( v ) (cid:1) ≥ L (cid:88) v ∈ V f ( v ) ≥ L (cid:88) x ∈ A f ( v ) = 2 L vol ( A ) > . HENG’S MAXIMAL DIAMETER THEOREM FOR HYPERGRAPHS 7
On the other hand (cid:88) v ∈ V (cid:0) L ρ p ( v ) + L ρ q ( v ) (cid:1) = (cid:10) L ρ p , D (cid:11) + (cid:10) L ρ q , D (cid:11) = 0 . This contradicts. Hence each point is on a geodesic path connecting from p to q .Take a pair of points v, w on a same geodesic path γ connecting from p to q .With out loss of generality, we may assume v = γ k , w = γ l with k < l . Note that (cid:104) ρ p , δ w − δ v (cid:105) = d ( v, w ) , since v, w are on γ . By (2.2), we have Kd ( p, v ) ≤ (cid:10) L ρ p , δ v − δ p (cid:11) ,Kd ( v, w ) ≤ (cid:10) L ρ p , δ w − δ v (cid:11) ,Kd ( w, q ) ≤ (cid:10) L ρ p , δ q − δ w (cid:11) . Adding these inequalities, we have Kd ( p, q ) = K ( d ( p, v ) + d ( v, w ) + d ( w, q )) ≤ (cid:10) L ρ p , δ q − δ p (cid:11) = Kd ( p, q ) . Hence all inequalities are actually equalities. Therefore K ≤ κ ( v, w ) ≤ κ ( v, w ) ≤ (cid:10) L ρ p , δ w − δ v (cid:11) d ( v, w ) = K. (cid:3) On Riemannian manifolds, Lichnerowicz-Obata’s theorem is also known. Itstates that the first non-zero eigenvalue λ of the Laplacian on n -dimensional Rie-mannian manifold M with Ric g ≥ ( n −
1) satisfies λ ≥ n and λ = n if and onlyif M = S n (1). We have a weaker theorem, whose proof is similar to [32]. Corollary 3.2.
Under the same assumption as in Theorem 3.1, then λ = K .Proof. We have λ ≥ K by Theorem 5.1 in [15]. By the proof of Theorem 3.1,inequalities (3.2) and (3.3) are actually equalities, that is, Kρ p ( v ) = L ρ p ( v ) + d v .Set f := ρ p − K − D L ( ρ p − K − D L ρ p (Lemma 2.3 in [15]), then L f ( v ) = L ( ρ p − K − D v ) = L ( ρ p )( v ) = Kρ p ( v ) − d v = Kf ( v ) . Hence λ = K . (cid:3) Remark . Since (3.3) is also equality, ρ q − K − D
1l is also an eigenfunction for λ = K . However due to the nonlinearity of L , the set of eigenfunctions is not alinear space. 4. Examples
Example 4.1.
Let V := { p, q, v , v } be a vertex set and E := { e i } i =1 a hyperedgeset, where e = pv , e = pv , e = v v , e = v q, e = v q, e = pv v , e = v v q. Define a weight function ω : E → R > by ω e ≡
1. Then the weighted hypergraph H = ( V, E , ω ) satisfies inf x (cid:54) = y κ ( x, y ) ≥
1, and
Diam V = 2. Thus H satisfiesthe assumption of Theorem 3.1. Actually v , v is on a geodesic from p to q , and κ ( p, v i ) = κ ( v i , q ) = 1 is satisfied. Example 4.2.
Let V := { p, q, v , v } be a vertex set and E := { e , e } a hyperedgeset, where e = pv v , e = v v q. YU KITABEPPU AND ERINA MATSUMOTO
Figure 1. H = ( V, E ) Define a weight function ω : E → R > by ω e ≡
1. Then the weighted hypergraph H = ( V, E , ω ) satisfies inf x (cid:54) = y κ ( x, y ) ≥
1, and
Diam V = 2. Thus H satisfiesthe assumption of Theorem 3.1. Actually v , v is on a geodesic from p to q , and κ ( p, v i ) = κ ( v i , q ) = 1 is satisfied. Figure 2. H = ( V, E ) Acknowledgement
The authors would like to thank Professors Shin-ichi Ohta and Yohei Sakuraifor their helpful comments and fruitful discussions. The first author is supportedby Grant-in-Aid for Young Scientists JP18K13412.
References [1] L. Ambrosio, N. Gigli, and G. Savar´e,
Bakry- ´Emery curvature-dimension condition and Rie-mannian Ricci curvature bounds , Ann. Probab. (2015), no. 1, 339–404, DOI 10.1214/14-AOP907. MR3298475[2] L. Ambrosio, N. Gigli, and G. Savar´e, Calculus and heat flow in metric measure spaces andapplications to spaces with Ricci bounds from below , Invent. Math. (2014), no. 2, 289–391,DOI 10.1007/s00222-013-0456-1. MR3152751
HENG’S MAXIMAL DIAMETER THEOREM FOR HYPERGRAPHS 9 [3] L. Ambrosio, N. Gigli, and G. Savar´e,
Metric measure spaces with Riemannian Ricci curva-ture bounded from below , Duke Math. J. (2014), no. 7, 1405–1490, DOI 10.1215/00127094-2681605. MR3205729[4] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala,
Riemannian Ricci curvature lower boundsin metric measure spaces with σ -finite measure , Trans. Amer. Math. Soc. (2015), no. 7,4661–4701, DOI 10.1090/S0002-9947-2015-06111-X. MR3335397[5] L. Ambrosio, A. Mondino, and G. Savar´e, Nonlinear diffusion equations and curvature con-ditions in metric measure spaces , Mem. Amer. Math. Soc. (2019), no. 1270, v+121, DOI10.1090/memo/1270. MR4044464[6] F. Bauer, P. Horn, Y. Lin, G. Lippner, D. Mangoubi, and S.-T. Yau,
Li-Yau inequality ongraphs , J. Differential Geom. (2015), no. 3, 359–405. MR3316971[7] D. Bakry and M. Ledoux, Sobolev inequalities and Myers’s diameter theorem for an abstractMarkov generator , Duke Math. J. (1996), no. 1, 253–270, DOI 10.1215/S0012-7094-96-08511-7. MR1412446[8] D. Bakry and Z. Qian, Volume comparison theorems without Jacobi fields , Current trendsin potential theory, Theta Ser. Adv. Math., vol. 4, Theta, Bucharest, 2005, pp. 115–122.MR2243959[9] F. Baudoin and N. Garofalo,
Curvature-dimension inequalities and Ricci lower bounds forsub-Riemannian manifolds with transverse symmetries , J. Eur. Math. Soc. (JEMS) (2017),no. 1, 151–219, DOI 10.4171/JEMS/663. MR3584561[10] A.-I. Bonciocat, A rough curvature-dimension condition for metric measure spaces , Cent.Eur. J. Math. (2014), no. 2, 362–380, DOI 10.2478/s11533-013-0332-7. MR3130690[11] A.-I. Bonciocat and K.-T. Sturm, Mass transportation and rough curvature bounds for dis-crete spaces , J. Funct. Anal. (2009), no. 9, 2944–2966, DOI 10.1016/j.jfa.2009.01.029.MR2502429[12] D. Cushing, S. Kamtue, J. Koolen, S. Liu, F. M¨unch, and N. Peyerimhoff,
Rigidity of theBonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature , Adv. Math. (2020), 107188, 53, DOI 10.1016/j.aim.2020.107188. MR4096132[13] M. Erbar, K. Kuwada, and K.-T. Sturm,
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces , Invent. Math. (2015), no. 3, 993–1071, DOI 10.1007/s00222-014-0563-7. MR3385639[14] M. Erbar and J. Maas,
Ricci curvature of finite Markov chains via convexity of the entropy ,Arch. Ration. Mech. Anal. (2012), no. 3, 997–1038, DOI 10.1007/s00205-012-0554-z.MR2989449[15] M. Ikeda, Y. Kitabeppu, and Y. Takai,
Coarse Ricci curvature of hypergraphs and its gener-alization , arXiv:2102.00698.[16] M. Ikeda, A. Miyauchi, Y. Takai, and Y. Yoshida,
Finding Cheeger cuts in hypergraphs viaheat equation , arXiv:1809.04396.[17] C. Ketterer,
Cones over metric measure spaces and the maximal diameter theorem , J. Math.Pures Appl. (9) (2015), no. 5, 1228–1275, DOI 10.1016/j.matpur.2014.10.011 (English,with English and French summaries). MR3333056[18] C. Ketterer and T. Rajala,
Failure of topological rigidity results for the measure contractionproperty , Potential Analysis (2015), no. 3, 645–655.[19] K. Kuwada, A probabilistic approach to the maximal diameter theorem , Math. Nachr. (2013), no. 4, 374–378, DOI 10.1002/mana.201100330. MR3028781[20] X.-M. Li,
On extensions of Myers’ theorem , Bull. London Math. Soc. (1995), no. 4, 392–396, DOI 10.1112/blms/27.4.392. MR1335292[21] X.-M. Li and F.-Y. Wang, On the compactness of manifolds , Infin. Dimens. Anal. Quan-tum Probab. Relat. Top. (2003), no. suppl., 29–38, DOI 10.1142/S0219025703001249.MR2074765[22] Y. Lin, L. Lu, and S.-T. Yau, Ricci curvature of graphs , Tohoku Math. J. (2) (2011),no. 4, 605–627, DOI 10.2748/tmj/1325886283. MR2872958[23] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport , Ann.of Math. (2) (2009), no. 3, 903–991, DOI 10.4007/annals.2009.169.903. MR2480619(2010i:53068)[24] J. Maas,
Gradient flows of the entropy for finite Markov chains , J. Funct. Anal. (2011),no. 8, 2250–2292, DOI 10.1016/j.jfa.2011.06.009. MR2824578[25] I. Miyadera,
Nonlinear semigroups , Translations of Mathematical Monographs, vol. 109,American Mathematical Society, Providence, RI, 1992. Translated from the 1977 Japaneseoriginal by Choong Yun Cho. MR1192132[26] F. M¨unch,
Remarks on curvature dimension conditions on graphs , Calc. Var. Partial Differen-tial Equations (2017), no. 1, Paper No. 11, 8, DOI 10.1007/s00526-016-1104-6. MR3592766 [27] F. M¨unch, Li-Yau inequality on finite graphs via non-linear curvature dimension condi-tions , J. Math. Pures Appl. (9) (2018), 130–164, DOI 10.1016/j.matpur.2018.10.006.MR3906157[28] S.-i. Ohta,
On the measure contraction property of metric measure spaces , Comment. Math.Helv. (2007), no. 4, 805–828, DOI 10.4171/CMH/110. MR2341840 (2008j:53075)[29] S.-I. Ohta, Products, cones, and suspensions of spaces with the measure contraction property ,J. Lond. Math. Soc. (2) (2007), no. 1, 225–236, DOI 10.1112/jlms/jdm057. MR2351619[30] Y. Ollivier, Ricci curvature of Markov chains on metric spaces , J. Funct. Anal. (2009),no. 3, 810–864, DOI 10.1016/j.jfa.2008.11.001. MR2484937 (2010j:58081)[31] R. Ozawa, Y. Sakurai, and T. Yamada,
Geometric and spectral properties of directed graphsunder a lower Ricci curvature bound , Calc. Var. Partial Differential Equations (2020),no. 4, Paper No. 142, 39, DOI 10.1007/s00526-020-01809-2. MR4135639[32] R. Ozawa, Y. Sakurai, and T. Yamada, Maximal diameter theorem for directed graphs ofpositive Ricci curvature , arXiv:2011.00755.[33] Q.-h. Ruan,
Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature , Proc.Japan Acad. Ser. A Math. Sci. (2009), no. 6, 71–74, DOI 10.3792/pjaa.85.71. MR2532422[34] M. Schmuckenschl¨ager, Curvature of nonlocal Markov generators , Convex geometric analysis(Berkeley, CA, 1996), Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge,1999, pp. 189–197. MR1665591[35] K.-T. Sturm,
On the geometry of metric measure spaces. I , Acta Math. (2006), no. 1,65–131, DOI 10.1007/s11511-006-0002-8. MR2237206 (2007k:53051a)[36] K.-T. Sturm,
On the geometry of metric measure spaces. II , Acta Math. (2006), no. 1,133–177, DOI 10.1007/s11511-006-0003-7. MR2237207 (2007k:53051b)[37] Y. Yoshida,
Cheeger inequalities for submodular transformations , Proceedings of the Thirti-eth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2019,pp. 2582–2601, DOI 10.1137/1.9781611975482.160. MR3909629(Yu Kitabeppu)
Kumamoto University
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