Harmonic Functions On Manifolds Whose Large Sphere Are Small

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.


INTRODUCTION
When (M, g) is a complete Riemannian manifold with non negative Ricci curvature, S-Y. Cheng and S-T. Yau have proven that any harmonic function h : M → R satisfies the gradient estimate [4] : A celebrated conjecture of S-T. Yau predicted the finite dimensionality of the space of harmonic functions with polynomial growth on a complete Riemannian manifold with non negative Ricci curvature : This conjecture has been proven by T. Colding and B. Minicozzi in a much more general setting. We say that a complete Riemannian manifold (M n , g) satisfies the doubling condition if there is a constant ϑ such that for any x ∈ M and radius R > 0 : vol B(x, 2R) ≤ ϑ vol B(x, R).
If B ⊂ M is a geodesic ball, we will use the notation r(B) for the radius of B and κB for the ball concentric to B and with radius κr(B). And if f is an integrable function on a subset Ω ⊂ M , we will note f Ω its mean over Ω: We say that a complete Riemannian manifold (M n , g) satisfies the scale (L 2 ) Poincaré inequality if there is a constant µ such that for any ball B ⊂ M and any function ϕ ∈ C 1 (2B): . Theorem. [5] If (M, g) is a complete Riemannian manifold that is doubling and that satisfies the scale Poincaré inequality then for any ν, the space of harmonic function of polynomial growth of order ν has finite dimension: It is well known that a complete Riemannian manifold with non negative Ricci curvature is doubling and satisfies the scale Poincaré inequality, hence the Yau's conjecture is true.
The proof is quantitative and gives a precise estimation of the dimension of dim H ν (M, g). In fact, the condition on the Poincaré inequality can be weakened and the result holds on a doubling manifold (M, g) that satisfies the mean value estimation [6,10] : for any harmonic function defined over a geodesic ball 3B : |h|.
An example of Riemannian manifold satisfying the above condition are Riemannian manifold (M, g) that outside a compact set (M, g) is isometric to the warped product ([1, ∞) × Σ, (dr) 2 + r 2γ h) where (Σ, h) is a closed connected manifold and γ ∈ (0, 1]. But when γ ∈ (0, 1), a direct analysis, separation of variables, shows that any harmonic function h satisfying for some ǫ > 0: h(x) = O e Cr 1−γ−ǫ is necessary constant. In particular, a harmonic function with polynomial growth is constant. In [8,9], A. Kasue has shown that this was a general result for manifold whose Ricci curvature satisfies a quadratic decay lower bound and whose geodesic spheres have sublinear growth (see also [11] for a related results): Theorem. If (M, g) is complete Riemannian manifold with a based point o whose Ricci curvature satisfies a quadratic decay lower bound: and whose geodeosic sphere have sublinear growth: then any harmonic function with polynomial growth is constant.
Following A. Grigor'yan and L. Saloff-Coste [7], we say that a ball B(x, r) is Our first main result is a refinement of A. Kasue's result when the hypothesis of the Ricci curvature is replaced by a scale Poincaré inequality for remote ball : There is a constant µ such that all remote balls B = B(x, r) satisfy a scale Poincaré inequality : Theorem A. Let (M, g) be a complete Riemannian manifold whose remote balls satisfy the scale Poincaré inequality and assume that geodesic spheres have sublinear growth: For instance, on such a manifold, a harmonic function h : M → R satisfying : for some γ ∈ (0, 1) then if h : M → R is a harmonic function such that for some positive constant C and ǫ: then h is constant. A by product of the proof will imply that on the class of manifold considered by A. Kasue, the doubling condition implies an estimateà la Cheng-Yau for for the gradient of harmonic function: Theorem B. Let (M n , g) be a complete Riemannian manifold that is doubling and whose Ricci curvature satisfies a quadratic decay lower bound. Assume that the diameter of geodesic sphere has a sublinear growth then there is a constant C such that for any geodesic ball B ⊂ M and any harmonic function h : This result has consequences for the boundness of the Riesz transform. When (M n , g) is a complete Riemannian manifold with infinite volume, the Green formula and the spectral theorem yield the equality: Hence the Riesz transform is a bounded operator. It is well known [12] that on a Euclidean space, the Riesz transform has a bounded extension R : L p (R n ) → L p (T * R n ) for every p ∈ (1, +∞). Also according to D. Bakry, the same is true on manifolds with non-negative Ricci curvature [2]. As it was noticed in [3, section 5], in the setting of the Theorem B, the analysis of A. Grigor'yan and L. Saloff-Coste [7] implies a scale L 1 -Poincaré inequality: there is a constant C such that any balls B = B(x, r) satisfies : And according to the analysis of P. Auscher and T. Coulhon   Proof. Let r ∈ [R + 4εR, 2R − 4εR], our hypothesis implies that there is some x ∈ ∂B(o, r) such that Let h : B(o, 2R) → R be a harmonic function and c ∈ R a real number. We use the Lipschitz function : Then integrating by part and using the fact that h is harmonic we get So that we have : and hence The hypothesis that ε ≤ 1/12 implies that the ball B(x, εR + εr) is remote, hence if we choose then the Poincaré inequality and the fact that r + R ≤ 3R imply : But we have : And for all r ∈ [R, R − 12εR] we get : We iterate this inequality and get provide that N 12εR ≤ R ; hence the result with C = 1 + 1 9µ and κ = 9µ 1 + 9µ 1 12 .

Harmonic function with polynomial growth.
We can now prove the following extension of Kasue's results : Theorem 2.2. Let (M, g) be a complete Riemannian manifold whose all remote balls B = B(x, r) satisfy a scale Poincaré inequality : Proof. Let h : M → R be a harmonic function with polynomial growth : We will defined We remark first that using the cut off function ξ defined by We obtain If we iterate the inequality obtained in Lemma 2.1, we get for all R such that ǫ(R) ≤ 1/12 : Using the estimation (1), we get But the Cesaro theorem convergence implies that : hence if we let ℓ → +∞ in the inequality (2) we get E R = 0 and this for all sufficiently large R, hence h is constant.

Extension.
A slight variation of the arguments yields the following extension: Theorem 2.3. Let (M, g) be a complete Riemannian manifold whose all remote balls B = B(x, r) satisfies a scale Poincaré inequality : Assume that the geodesic spheres have sublinear diameter growth : Proof. Indeed, the above argumentation shows that if R is large enough then .
Hence we get the inequality : It is then easy to conclude.

LIPSCHITZ REGULARITY OF HARMONIC FUNCTIONS
We are going to prove that a Lipschitz regularity for harmonic function analogous to the the Cheng-Yau gradient inequality : Hence the estimate (3) holds with C = max{B, ϑ}.