Spectrally positive Bakry-Émery Ricci curvature on graphs

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Abstract

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that L+2Ric is a positive operator where L is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small in Kato sense, we prove diameter bounds, elliptic Harnack inequality and Buser inequality. This article seems to be the first one establishing these results while allowing for some negative curvature.

Résumé

Nous étudions les implications analytiques et géométriques des limites de courbure de Ricci non constantes. Nous prouvons une estimation de Lichnerowicz des valeurs propres et la finitude du groupe fondamental en supposant que L+2Ric est un opérateur positif où L est le graphe laplacien. En supposant que la partie négative de la courbure de Ricci est petite dans le sens de Kato, nous prouvons des bornes de diamètre, l'inégalité elliptique de Harnack et l'inégalité de Buser. Cet article semble être le premier à établir ces résultats tout en permettant une certaine courbure négative.

Introduction

Analytic properties of graphs and manifolds such as eigenvalue estimates for the Laplacian or gradient estimates for the heat semigroup depend crucially on curvature conditions. In the last decade, there has been increasing interest in discrete Ricci curvature notions based on [25], [19], [6], [16]. Indeed, most of the research so far is focused on constant curvature bounds.

Recently, authors began to generalize the uniform bounds in the curvature dimension condition to obtain Bonnet-Myers type theorems under almost positivity assumptions with exceptions [14], [17]. A big gap in dealing with almost positive curvature is that it is not clear that Sobolev inequalities hold on graphs even in the case of uniformly positive curvature, such that techniques based on heat kernel estimates do not carry over immediately. To overcome this issue, we deal here with spectrally positive Ricci curvature in the sense that for some K0, we have12L+ρK, where L=Δ0 is the non-negative Laplacian of the graph and ρ:VR maps every point to the smallest value of the Bakry-Émery Ricci curvature at a given vertex. This will be made precise in the next section.

Spectrally positive Ricci curvature on Riemannian manifolds proved to be a fruitful analytic generalization for most of the well-known results depending on uniformly bounded Ricci curvature or smallness of its negative part in Lp-sense, see [4]. A very powerful assumption implying spectral positivity is the so-called Kato condition [22], [2], [20], [24], [21]. Denote for xR its negative part by x:=min{0,x}. The function W0 satisfies the Kato condition if there is a T>0 such thatkT(W):=0TPtWdt<1, where (Pt)t0 denotes the heat semigroup. The major advantage of this condition is that one can treat mapping properties of perturbed semigroups without knowing anything about the perturbed heat kernel itself, [23], allowing to obtain explicit results as soon as one knows something about the unperturbed heat kernel. Here, we use this condition for the function W:=(ρK) for some K>0.

In this paper, we will prove Lichnerowicz estimates for Bakry-Émery spectrally positive Ricci curvature, and we prove a Bonnet-Myers type theorem under a Kato condition on the curvature. Therefore, we use a combination of the original proof of Lichnerowicz together with the Perron-Frobenius theorem. Also under spectrally positive Ricci curvature, we prove finiteness of the fundamental group. Assuming additionally that a refinement of (2) holds, we prove a gradient estimate for the perturbed semigroup generated by Δ+2ρ, and use perturbation results to get elliptic Harnack and Buser inequalities. Originally, Harnack and Buser inequality are stated for non-negatively curvature. Our version requires a Kato condition comparing to positive curvature as some positive curvature is required to compensate the admissible occurrence of negative curvature.

Section snippets

Graph Laplacians and variable Bakry-Émery curvature

A measured and weighted graph (short mwg) G=(V,w,m) is a triple consisting of a countable set V, a symmetric function w:V×V[0,) which is zero on the diagonal, and a function m:V(0,). We always assume that mwg are locally finite, i.e., given G=(V,w,m) and xV, we always assume that the set of yV with w(x,y)>0 is finite, and that G is connected. We abbreviate q(x,y):=w(x,y)/m(x), and define Deg(x):=yVq(x,y), xV. Furthermore, we let Degmax:=supxDeg(x).

Definition 2.1

Let G=(V,w,m) be a mwg. We define the

Lichnerowicz estimate for spectrally positive curvature

We first prove a Lichnerowicz-type theorem assuming spectrally positive Ricci curvature since it does not depend on the Kato condition.

Theorem 3.1

Let K>0, ρC(V), n(0,], and G a (K,n,ρ)-spectrally positive finite mwg. Then, we have λ(G)K, where λ(G) is the first positive eigenvalue of L in L2(V).

Proof

Let f be an eigenfunction to the eigenvalue λ of L, i.e., Lf=λf and let ϕ0. Then,12Δϕ,Γf+λϕ,Γf=12ϕ,ΔΓfϕ,Γ(f,Δf)=Γ2f,ϕΓf,ρϕ and thus,λϕ,Γf(L/2+ρ)ϕ,Γf. Let E be the smallest eigenvalue of L/2+ρ

Gradient estimates and Harnack inequality

A modification of the classical Ledoux ansatz leads to the following gradient estimate.

Theorem 4.1

Let G be a mwg with Degmax< satisfying CD(ρ,n) for ρ:VR and n(0,]. Then, for all bounded fC(V), we have the pointwise inequalities

  • (i)

    ΓPtfPt2ρΓf,

  • (ii)

    2tΓPtfPt2ρf2(Ptf)2,

  • (iii)

    and assuming ρK, we get12K(e2Kt1)ΓPtfe2KtPt2ρf2(Ptf)2.

Proof

As usual, we omit the dependency of the vertex. Define for fixed fC(V)H(s):=Ps2ρΓPtsf. Then, following the classical Ledoux ansatz, we have H(s)0, where we used the fact that the

Diameter bounds and curvature in Kato class

Similarly to [12], [14], we apply a gradient estimate (Theorem 4.1) to conclude a diameter bound.

Theorem 5.1

Let T,K>0, n(0,], ρC(V), G a mwg with Degmax< and CD(ρ,n). SupposekT((ρK))14(1eKT/2). Then,diam(G)4DegmaxeKT/2/K.

Proof

We follow the proof of [12, Theorem 2.1]. First, note that for any gC(V), we have (Δg(x))22Deg(x)Γg(x), xV. Fix yV, R>0, and define f(x):=(d(x,y)R)+. Thanks to Γf1, and (3), we have for all t>0 and all xV,|tPtf(x)|2=|ΔPtf(x)|22Deg(x)ΓPtf(x)2DegmaxPt2ρΓf(x)Degmax2eKTeK

Fundamental group and spectrally positive curvature

It has been shown in [9] that strictly positive Bakry Emery curvature everywhere implies finiteness of the fundamental group. By the fundamental group we refer to the CW-complex consisting of the graph as 1-skeleton where all cycles of length 3 and 4 are filled with disks. In order to obtain finiteness of the fundamental group, it turns out to be sufficient to assume spectrally positive curvature which is significantly weaker than the Kato condition or a uniform positive lower curvature bound.

Theorem 6.1

Buser's inequality and curvature in Kato class

We prove a Buser inequality under our generalized curvature conditions. Therefore, we need to introduce the Cheeger constant of a finite weighted graph G, that is given byh(G):=inf{|U|volU|UV,volU12volV}, where |U|:=xU,yVUw(x,y) and volU=xUm(x).

Buser's inequality gives the counterpart to the well known Cheeger inequality stating that the first positive eigenvalue is larger than the square of the Cheeger constant h up to a constant factor. Buser inequality is the reverse inequality

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