Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II

In a previous paper Hua-Jost-Liu, we have applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincar\'e inequality on these graphs to obtain an dimension estimate of polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on graphs, we directly argue on Alexandrov spaces to obtain the optimal dimension estimate of polynomial growth harmonic functions on graphs which is actually linear in the growth rate. From a harmonic function on the graph, we construct a function on the corresponing Alexandrov surface that is not necessarily harmonic, but satisfies crucial estimates.


Introduction
This paper is the second one in a series studying geometric analysis aspects of infinite graphs with nonnegative curvature.We refine the argument in Hua-Jost-Liu [27] and introduce a new observation to obtain the asymptotically optimal dimension estimate of the space of polynomial growth harmonic functions on such graphs.
In 1975, Yau [50] proved the Liouville theorem for harmonic functions on Riemannian manifolds with nonnegative Ricci curvature.Soon after Cheng-Yau [8] obtained the gradient estimate for positive harmonic functions which implies that sublinear growth harmonic functions on these manifolds are constant.Then Yau [51,52] conjectured that the space of polynomial growth harmonic functions with growth rate less than or equal to d on Riemannian manifolds with nonnegative Ricci curvature is of finite dimension.Li-Tam [38] and Donnelly-Fefferman [19] independently solved the conjecture for 2-dimensional manifolds.Then Colding-Minicozzi [10,11,12] gave the affirmative answer for any dimension by using the volume comparison property and the Poincaré inequality.Later, Li [36] and Colding-Minicozzi [13] simplified the proof by the mean value inequality.The dimension estimates in [12,13,36] are asymptotically optimal.In the wake of this result, many generalizations on manifolds [48,47,46,39,40,41,7,29,35] and on singular spaces [15,33,25,26] followed.In this paper, we obtain the optimal dimension estimate which is linear in d rather than quadratic in d as in [27].
Let us now describe the results in more details.The combinatorial curvature for planar graphs was introduced by [44,21,28] and studied by many authors [23,49,17,6,5,45,43,3,4,30,31,32].Let G = (V, E, F ) be a (called semiplanar) graph embedded in a 2-manifold such that each face is homeomorphic to a closed disk with finite edges as the boundary.Let S(G) be the regular polygonal surface obtained by assigning length one to every edge and filling regular polygons in the faces of G.The combinatorial curvature at the vertex x is defined as , where d x is the degree of the vertex x, deg(σ) is the degree of the face σ, and the sum is taken over all faces incident to x (i.e.x ∈ σ).The idea of this definition is to measure the difference of 2π and the total angle Σ x at the vertex x on the regular polygonal surface S(G) equipped with a metric structure obtained from replacing each face of G with a regular polygon of side length one and gluing them along the common edges.That is, 2πΦ(x) = 2π − Σ x .
It was proved in [27] that G has nonnegative combinatorial curvature everywhere if and only if the corresponding regular polygonal surface S(G) is an Alexandrov space with nonnegative sectional curvature, i.e.SecS(G) ≥ 0 (or SecG ≥ 0 for short).This class of graphs includes all regular tilings of the plane (see [22]) and more general graphs (see [5,27]).
For the basic facts of Alexandrov spaces, readers are referred to [2,1].In this paper, we only consider 2-dimensional Alexandrov spaces with nonnegative curvature (in particular convex surfaces).Let G be a semiplanar graph with SecG ≥ 0, then X := S(G) is a 2-dimensional Alexandrov space with nonnegative curvature.We denote by d the intrinsic metric on X, by B R (p) := {x ∈ X | d(x, p) ≤ R} the closed geodesic ball on X, and by The well known Bishop-Gromov volume comparison holds on X (see [1]) that for any p ∈ X, 0 < r < R, we have We call (1.1) the relative volume comparison and (1.2) the volume doubling property.The Poincaré inequality was proved in [34,25] on Alexandrov spaces.For any p ∈ X, R > 0 and any Lipschitz function u on X, where u BR = 1 |BR(p)| ´BR(p) u, and |▽u| is the a.e.defined norm of the gradient of u.
It has been shown in [27] that G inherits some geometric estimates from those of X := S(G).[6].Then the weak relative volume comparison (1.4) and the volume doubling property (1.5) were obtained in [27] for SecG ≥ 0. For any p ∈ G, 0 < r < R, ) where C(D) are constants only depending on D. The Poincaré inequality on G was also obtained in [27].There exist two constants C(D) and C such that for where x , and x ∼ y means that x and y are neighbors in G.
A function f on G is called discrete harmonic (see [20,18,9]) if for ∀x ∈ G, Let G be a semiplanar graph with nonnegative curvature and  [36] and Colding-Minicozzi [13] obtained the optimal dimension estimate by the mean value inequality.In the graph case, the volume doubling property (1.5) and the Poincaré inequality (1.6) imply that dim D) where C(D) and v(D) depending on the maximal facial degree D (see [15]).Hua-Jost-Liu [27] used the weak relative volume comparison (1.4) to obtain the estimate dim It is obviously not optimal.But it is hard to obtain the optimal dimension estimate on the graph G since the constant!C(D) in the weak relative volume comparison (1.4) may not be close to 1.
Here comes the key observation.Since the relative volume comparison (1.1) on X is as nice as in the case of Riemannian manifolds, we do the dimension estimate argument for H d (G) on X.For any (discrete) harmonic function f on G, we extend it to a function f defined on X with controlled behavior (see (3.3)(3.4)).But in general, the extended function f may not be harmonic on X anymore, nor will f 2 be subharmonic.However, since the original harmonic function f satisfies the mean value inequality on G (see Lemma 3.2), the extended function f satisfies the mean value inequality in the large.
Theorem 1.1 (Mean value inequality on X).Let G be a semiplanar graph with SecG ≥ 0. Then there exist constants R 1 (D), C 2 (D) such that for any p ∈ X, R ≥ R 1 (D) and any harmonic function f on G we have (1.7) d(p, x) + 1) d } denote the space of polynomial growth functions on X with growth rate less than or equal to d.Since the extending map is an injective linear operator, it suffices to get the dimension estimate for the image E(H d (G)).Combining the relative volume comparison (1.1) and the mean value inequality (1.7), we obtain the optimal dimension estimate for E(H d (G)).
Although we have to pay for extending map E by the loss of harmonicity, it preserves the mean value property which is sufficient for our application.We adopt the argument of the mean value inequality (see [36,37,13]) to get optimal dimension estimate.In addition, by the special structure of the graph with SecG ≥ 0 and D ≥ 43, we [27] obtained that for any d > 0, dim H d (G) = 1, which implies the final theorem of the paper.
Theorem 1.2.Let G be a semiplanar graph with SecG ≥ 0. Then for any where C is an absolute constant.
From a superficial glance, it might look as if polynomial growth harmonic functions on Riemannian manifolds (continuous objects) and those on graphs (discrete ones) are very similar and might succumb to an analogous treatment.While our work is indeed inspired by certain analogies, there are also some important differences which necessitate new ideas which we now wish to summarize.Firstly, the unique continuation property for (discrete) harmonic functions on graphs fails, leaving us with the problem of verifying the inner product property of the bilinear form L 2 (B R ) on H d (G) where B R is the geodesic ball of radius R in a graph G (see (4.1)).We use a lemma in [26] (see Lemma 4.1 in this paper) to overcome this difficulty.Secondly, the constant C(D) in the relative volume comparison (1.4) on semiplanar graphs with nonnegative curvature is not necessarily close to 1.Even on manifolds, it is still an open problem to obtain the optimal dimension estimate by using (1.4) and (1.6).In this paper, we find an argument which transforms the discrete harmonic functions on the semiplanar graph G with nonnegative curvature to functions on the polygonal surface S(G) that satisfy the mean value inequality.This crucial step enables us to transfer the argument to S(G) where we have a nice volume comparison (1.1) and to obtain the optimal dimension estimate of H d (G).Thirdly, the combinatorial obstruction for semiplanar graphs with a large face (i.e.D ≥ 43) makes the dimension estimate independent of the parameter D.

Preliminaries and Notations
We recall the definition of semiplanar graphs in [27].
be embedded into a connected 2-manifold S without self-intersections of edges and such that each face is homeomorphic to the closed disk with finite edges as the boundary.
Let G = (V, E, F ) denote the semiplanar graph with the set of vertices, V , edges, E and faces, F .Edges and faces are regarded as closed subsets of S, and two objects from V, E, F are called incident if one is a proper subset of the other.We always assume that the surface S has no boundary and the graph G is a simple graph, i.e. without loops and multi-edges.We denote by d x the degree of the vertex x ∈ G and by deg(σ) the degree of the face σ ∈ F , i.e. the number of edges incident to σ.Further, we assume that 3 ≤ d x < ∞ and 3 ≤ deg(σ) < ∞ for each vertex x and face σ, which means that G is a locally finite graph.For each semiplanar graph G = (V, E, F ), there is a unique metric space, denoted by S(G), which is obtained from replacing each face of G by a regular polygon of side length one with the same facial degree and gluing the faces along the common edges in S. S(G) is called the regular polygonal surface of the semiplanar graph G.
For a semiplanar graph G, the combinatorial curvature at each vertex x ∈ G is defined as , where the sum is taken over all the faces incident to x.In this paper, we only consider semiplanar graphs with nonnegative combinatorial curvature.It was proved in [27] that a semiplanar graph G has nonnegative combinatorial curvature everywhere if and only if the regular polygonal surface S(G) is an Alexandrov space with nonnegative curvature, denoted by SecG ≥ 0 or SecS(G) ≥ 0.
For Alexandrov spaces and Alexandrov geometry, readers are referred to The length of a curve γ is defined as A curve γ is called rectifiable if L(γ) < ∞.Given x, y ∈ X, denote by Γ(x, y) the set of rectifiable curves joining x and y.A metric space (X, d) is called a length space if d(x, y) = inf γ∈Γ(x,y) {L(γ)}, for any x, y ∈ X, where d is called the intrinsic metric on X.
It is always true by the definition of the length of a curve that d(γ(a), γ(b)) ≤ L(γ).A geodesic is a shortest curve (or shortest path) joining the two end points.A geodesic space is a length space (X, d) satisfying that for any x, y ∈ X, there is a geodesic joining x and y.
Denote by Π κ , κ ∈ R the model space which is a 2-dimensional, simply connected space form of constant curvature κ.Typical ones are In a geodesic space (X, d), we denote by γ xy one of the geodesics joining x and y, for x, y ∈ X.Given three points x, y, z ∈ X, denote by △ xyz the geodesic triangle with edges γ xy , γ yz , γ zx .There exists a unique (up to an isometry) geodesic triangle, In other words, an Alexandrov space (X, d) is a geodesic space which locally satisfies the Toponogov triangle comparison theorem for the sectional curvature.It was proved in [2] that the Hausdorff dimension of an Alexandrov space (X, d), dim H (X), is an integer or infinity.In this paper, we only consider 2-dimensional Alexandrov spaces with SecX ≥ 0.
Let G be a semiplanar graph with nonnegative combinatorial curvature.Let X := S(G) be the regular polygonal surface of G with the intrinsic metric d.Then SecX ≥ 0. Let B R (p) denote the closed geodesic ball centered at p ∈ X of radius R > 0, i. Lemma 2.3.Let (X, d) be an 2-dimensional Alexandrov space with nonnegative curvature, i.e.SecX ≥ 0. Then for any p ∈ X, 0 < r < R, it holds that (2.1) We call (2.1) the relative volume comparison and (2.2) the volume doubling property.
For any precompact domain Ω ⊂ X, we denote by Lip(Ω) the set of Lipschitz functions on Ω.For any f ∈ Lip(Ω), the W 1,2 norm of f is defined as The W 1,2 space on Ω, denoted by W 1,2 (Ω), is the completion of Lip(Ω) with respect to the W 1,2 norm.A function f ∈ W 1,2 loc (X) if for any precompact domain Ω ⊂⊂ X, f | Ω ∈ W 1,2 (Ω).The Poincaré inequality was proved in [34,25]. ) ) be a semiplanar graph with nonnegative combinatorial curvature and X := S(G) be the regular polygonal surface of G. Then it is straightforward that 3 ≤ d x ≤ 6 for ∀x ∈ G, i.e.G has bounded degree.We denote by D := D G := sup{deg(σ) : σ ∈ F } the maximal degree of faces in G, which is a very important parameter in our discussion (it is finite by Gauss-Bonnet formula in [17,6]).For any x, y ∈ G, they are called neighbors, denoted by x ∼ y, if there is an edge in E connecting x and y.There is a natural metric on the graph G, d G (x, y) := inf{k : ∃x = x 0 ∼ • • • ∼ x k = y}, i.e. the length of the shortest path connecting x and y by assigning each edge the length one.Lemma 3.1 in [27] implies that the two metrics, d G and d, on G are bi-Lipschitz equivalent, i.e. there exists a universal constant C such that for any x, y ∈ G For any p ∈ G and R > 0, we denote by Lemma 2.5.Let G = (V, E, F ) be a semiplanar graph with SecG ≥ 0. Then there exists a constant C(D) depending on D, such that for any p ∈ G and 0 < r < R, we have We call (2.5) the weak relative volume comparison and (2.6) the volume doubling property on G.The Poincaré inequality on G was also obtained in [27] by the Poincaré inequality (2.3).Lemma 2.6.Let G be a semiplanar graph with SecG ≥ 0. Then there exist two constants C(D) and C such that for any p ∈ G, R > 0, f : where

Mean Value Inequality
In this section, we extend each harmonic function on the semiplanar graph G with nonnegative combinatorial curvature to a function on X := S(G) which is almost harmonic in the sense that it satisfies the mean value inequality on X.
For any Ω ⊂ G and x ∈ G, we define d G (x, Ω) := inf{d G (x, y) | y ∈ Ω}.We denote for any x ∈ Ω, where L is called the Laplacian operator.
Since the volume doubling property (2.6) and the Poincaré inequality (2.7) are obtained on the semiplanar graph with nonnegative combinatorial curvature, the Moser iteration can be carried out (see [16,24]).

Lemma 3.1 (Harnack inequality).
Let G be a semiplanar graph with SecG ≥ 0. Then there exist constants C 1 (D) and C 2 (D) such that for any p ∈ G, R ≥ 1 and any positive harmonic function f on B G C1R (p) we have max The mean value inequality is one part of the Moser iteration (see also [14]).

Lemma 3.2 (Mean value inequality on graphs).
Let G be a semiplanar graph with SecG ≥ 0. Then there exist two constants C 1 (D) and C 2 (D) such that for any R > 0, p ∈ G, any harmonic function f on B G C1R (p), we have In the following process, we extend each function defined on G to the function f defined on X := S(G) with controlled behavior.Let f be a function on G, f : G → R, G 1 be the 1-dimensional simplicial complex of G by assigning each edge the length one.Step one is the linear interpolation, i.e. f is extended to a piecewise linear function on G 1 , f 1 : G 1 → R. In step two, we extend f 1 to a function defined on each face of G.For any regular n-polygon △ n of side length one, there is a bi-Lipschitz map where B rn is the circumscribed circle of △ n of radius r n = 1 2 sin αn 2 (for α n = 2π n ).Without loss of generality, we may assume that the origin o = (0, 0) of R 2 is the barycenter of △ n , the point (x, y) = (r n , 0) ∈ R 2 is a vertex of △ n , and B rn = B ! r n (o).Then in polar coordinates, L n reads where Brn (o) is the open disk.Then we define f : for any σ ∈ F. It is easy to see that f is continuous function (actually it is in W 1,2 loc (X)).
We improve the estimates in [27] to control the behavior of f .Let B 1 be the closed unit disk in R 2 .For completeness, we give the proof here. where So the harmonic function g with boundary value h is Since ∆g = 0, we have ∆g 2 = 2|∇g| 2 , then which follows from integration by parts.So that In addition, Hence, The second part of the theorem follows from an integration by parts and the Hölder inequality.
(by |x| ≤ 1) Then it holds that where ǫ is small, T = 1 rn ∂ θ is the unit tangent vector on the boundary ∂B rn and h T is the directional derivative of h in T.
The following lemma follows from the bi-Lipschitz property of the map L n : △ n → B rn .Lemma 3.5.Let G be a semiplanar graph with SecG ≥ 0 and σ := △ n .Then we have Proof.By the bi-Lipschitz property of L n and the inequality (3.6), we have where T n is the unit tangent vector on the boundary ∂△ n .Let e ⊂ △ n be an edge with two incident vertices, u and v.By linear interpolation, we have In addition, Hence, by (3.8) (3.9) and (3.10), we have By setting ǫ = Let G = (V, E, F ) be a semiplanar graph with SecG ≥ 0. For any p ∈ X, there exists a face σ ∈ F such that p ∈ σ.For any vertex q ∈ σ ∩ G, we have d(p, q) ≤ C 3 (D), since diamσ ≤ C 3 (D) for deg(σ) ≤ D. Note that 3 ≤ d x ≤ 6, for any x ∈ G. Lemma 3.6.Let G be a semiplanar graph with SecG ≥ 0. Then there exists a constant C(D) such that for any p ∈ X, q ∈ G on same face, we have where r > 2C3(D)
For any vertex x ∈ W r ′ ∩ G, there exists a face σ 1 ∈ W r ′ such that x ∈ σ 1 , so that Hence by (2.4) we have d G (q, x) ≤ r which implies that where ♯W r ′ is the number of faces in W r ′ .Moreover, where Hence the lemma follows from (3.14) (3.15) and (3.13), Now we can prove the mean value inequality for the extended function f defined on X := S(G) for some harmonic function f on G.
Proof of Theorem 1.1.For any p ∈ X, there exists a face △ n such that p ∈ △ n .Then by the construction of f (see (3.3) where the last inequality follows from the mean value inequality (3.2) for harmonic functions on the graph G.By (3.12) in Lemma 3.6, where the last inequality follows from the relative volume comparison (2.1) on X.

Optimal Dimension Estimate
In this section, we estimate the dimension of the space of polynomial growth harmonic functions on a semiplanar graph with nonnegative combinatorial curvature.
Let G be a semiplanar graph with SecG ≥ 0. For some fixed p ∈ G, we denote by But the optimal dimension estimate is linear in d as in the Riemannian case (see [12,13,36]).On the graph G, it is hard to obtain a nice relative volume comparison.But on the Alexandrov space X := S(G), the relative volume comparison (2.1) follows from the Bishop-Gromov volume comparison theorem.To obtain the asymptotically optimal dimension estimate, we argue on the Alexandrov space X instead of G.
We denote by the space of polynomial growth functions on X with growth rate less than or equal to d.For any harmonic function on G, we extend it to the function f defined on X in the process of (3.3) and (3.4) which establishes a map It is easy to see that E is an injective linear operator.Hence it suffices to get the dimension estimate of the image E(H d (G)).the relative volume comparison (2.1) on X and the mean value inequality (1.7) for each function in E(H d (G)), we obtain the optimal dimension estimate (see [36,37,13,26]).Lemma 4.1.For any finite dimensional subspace K ⊂ E(H d (G)), there exists a constant R 0 (K) depending on K such that for any R ≥ R 0 (K), is an inner product on K.
Lemma 4.2.Let G be a semiplanar graph with SecG ≥ 0, K be a k-dimensional The following lemma follows from the mean value inequality (1.7) for the extended functions.Since A p (t) = V ′ p (t) a.e., we integrate by parts and obtain Noting that f ′ (t) ≥ 0 and the relative volume comparison (2.1), we have Combining this with (4.3), we prove the lemma.
Proof of Theorem 1.2.For any k−dimensional subspace K ⊂ E(H d (G)), we set β = 1 + ǫ, for fixed small ǫ.By Lemma 4.2, there exists infinitely many > R 0 (K) such that for any orthonormal basis {u i } k i=1 of K with respect to A (1+ǫ)R , we have For any p ∈ G and R > 0, we denote by d G the distance on the graph G, by B G R (p) = {x ∈ G : d G (p, x) ≤ R} the closed geodesic ball on G and by |B G R (p)| := x∈B G R (p) d x the volume of B G R (p).Let D := D G denote the maximal degree of the faces in G, i.e.D := D G := sup σ∈F deg(σ) which is finite by y), d(ȳ, z) = d(y, z) and d(z, x) = d(z, x).We call △ xȳ z the comparison triangle in Π κ .Definition 2.2.A complete geodesic space (X, d) is called an Alexandrov space with sectional curvature bounded below by κ (SecX ≥ κ for short) if for any p ∈ X, there exists a neighborhood U p of p such that for any x, y, z ∈ U p , any geodesic triangle △ xyz , and any w ∈ γ yz , letting w ∈ γ ȳ z be in the comparison triangle △ xȳ z in Π κ satisfying d(ȳ, w) = d(y, w) and d( w, z) = d(w, z), we have d(x, w) ≥ d(x, w).
x) ≤ R} the closed geodesic ball in the graph G, by |B G R (p)| := x∈BR(p) d x the volume of B G R (p), and by ♯B G R (p) the number of vertices in the closed geodesic ball B G R (p).Since 3 ≤ d x ≤ 6 for any x ∈ G, |B G R (p)| and ♯B G R (p) are equivalent up to a constant, i.e. 3♯B G R (p) ≤ |B G R (p)| ≤ 6♯B G R (p), for any p ∈ G and R > 0. The following volume comparison on G was proved in [27] by the relative volume comparison (2.1) on X.

.
It maps the boundary of △ n to the boundary of B rn (o).Direct calculation shows that L n is a bi-Lipschitz map, i.e. for any x, y ∈ △ n we have C 1 |x − y| ≤ |L n x − L n y| ≤ C 2 |x − y|, where C 1 and C 2 do not depend on n.Then for any σ ∈ F, we denote σ := △ n where n := deg(σ).Let g : B rn (o) → R satisfy the following boundary value problem

Lemma 3 . 3 .
For any Lipschitz function h : ∂B 1 → R, let g : B 1 → R satisfy the following boundary value problem

) 1 √ 3 ≤ r n = 1
Note that for the semiplanar graph G with nonnegative curvature and any face σ = △ n of G, we have 3 ≤ n ≤ D, Then the scaled version of Lemma 3.3 reads Lemma 3.4.For 3 ≤ n ≤ D, and any Lipschitz function h : ∂B rn → R, we denote by g the harmonic function satisfying the Dirichlet boundary value problem ß ∆g = 0, in Brn g| ∂Br n = h .
the space of polynomial growth harmonic functions on G with growth rate less than or equal to d.By the method of Colding-Minicozzi, the volume doubling property (2.6) and the Poincaré inequality (2.7) imply that dim H d (G) ≤ C(D)d v(D) for d ≥ 1, where C(D) and v(D) are constants depending on the maximal facial degree D of G. Hua-Jost-Liu [27] used the weak relative volume comparison (2.5) on the graph G and the Poincaré inequality to obtain the dimension estimate dim H d (G) ≤ Cd 2 .