On the exponent of convergence of negatively curved manifolds without Green’s function

In this paper we prove that for every complete n-dimensional Riemannian manifold without Green’s function and with its sectional curvatures satisfying K ≤ −1, the exponent of convergence is greater than or equal to n− 1. Furthermore, we show that this inequality is sharp. This result is well known for manifolds with constant sectional curvatures K = −1.

In this paper we prove same relations between some useful concepts of Potential Theory: the first eigenvalue of the Laplace-Beltrami operator, the existence of Green's function and the exponent of convergence, for complete Riemannian n-manifolds (n ≥ 2) with sectional curvatures K ≤ −1. Note that the case of complete Riemannian n-manifolds with sectional curvatures K ≤ −k 2 < 0 can be reduced to this one.
These concepts of Potential Theory are related with other interesting topics as, for instance, heat kernel, isoperimetric inequalities, conic limit set and geodesics in the manifold (see Section 2).
Our main result is Theorem 3.7, which states that for every complete n-dimensional Riemannian manifold without Green's function and with sectional curvatures satisfying K ≤ −1, the exponent of convergence is greater than or equal to n − 1. This inequality is sharp, as we show after the proof of this theorem. Theorem 3.7 also states that the first eigenvalue of every complete n-dimensional Riemannian manifold without Green's function is 0 (without hypothesis on curvature). These results are well known for manifolds with constant sectional curvatures K = −1.
Note that to determine the first eigenvalue is not easy even if the manifold is negatively curved and its topology is trivial: if M is a simply connected complete n-dimensional Riemannian manifold with sectional curvatures satisfying K ≤ −1, we just know that λ 1 ≥ (n − 1) 2 /4 (see [15] and [20]).

Some background on Potential Theory.
In this section we define the concepts on Potential Theory that we need. We start with the fundamental tone (the first eigenvalue of the Laplace-Beltrami operator).
Definition 2.1. Let M be a Riemannian manifold and Ω ⊆ M be a domain in M . The fundamental tone λ 1 (Ω) of Ω is defined in terms of the Rayleigh's quotient as It is well known that if Ω is a normal domain (a domain with compact closure and smooth non-empty boundary), the fundamental tone λ 1 (Ω) is the lowest (first) eigenvalue of the positive Laplace-Beltrami operator ∆ := − div grad, with zero boundary value.
Eigenvalues of the Laplace-Beltrami operator play an important role in Riemmannian geometry (see, e.g., [8]). In particular, the fundamental tone of a manifold is related with its exponent of convergence (see Theorem 3.5) and isoperimetric inequalities: a general result of Cheeger says that λ 1 h 2 ≥ 1/4 for every complete Riemannian manifold with infinite volume, where h is the smallest linear isoperimetric constant of the manifold; it turns out that negative curvature forces an inequality in the opposite direction (see [6]); hence, to have the linear isoperimetric inequality (h < ∞) in a negatively curved manifold is equivalent to λ 1 > 0. The linear isoperimetric inequality is closely related to the project of Ancona on the space of positive harmonic functions of Gromov-hyperbolic manifolds (see [1], [2] and [3]); in particular, Cao proves that for a large class of Gromov-hyperbolic manifolds (and graphs) the linear isoperimetric inequality implies that the Dirichlet problem at infinity for the Laplace-Beltrami operator is solvable (see [7]). Isoperimetric constants are also related with the geometry of ends and large time heat diffusion in Riemannian manifolds (see [9]).
We recall that a Green's function in a complete Riemannian manifold M is a positive fundamental solution of the Laplace-Beltrami operator on M . It is well known that a complete manifold has Green's function if and only if there exists a non-constant positive superharmonic function (see, e.g., [21]). In terms of Brownian motion, a complete manifold has Green's function if and only if the Brownian motion on the manifold is transient (i.e., the Brownian motion eventually escapes from any compact set with probability 1).
There is also another useful characterization of the existence of Green's function: The harmonic measure of the ideal boundary of a complete Riemannian manifold with respect to a normal domain Ω 0 ⊂ M is defined as follows. Let Ω be another normal domain containing Ω 0 . We denote by u Ω the harmonic function in Ω \ Ω 0 which is 0 on ∂Ω 0 and 1 on ∂Ω \ ∂Ω 0 . The maximum principle implies that u Ω decreases when Ω increases. Hence the harmonic limit function is well defined in M \ Ω 0 . It is either identically 0 or positive and less than 1. In the first case we say that the harmonic measure of the ideal boundary vanishes; this is equivalent to It is easy to check that this definition does not depend on the choice of Ω 0 . Green's function is also related with other interesting topics as the heat kernel and isoperimetric inequalities: it is well known that Green's function is the integral of the heat kernel of the manifold; Fernández proves in [10] that the existence of some kind of isoperimetric inequalities guarantees the existence of Green's function for Riemannian manifolds; the results in [13] imply that the linear isoperimetric inequality guarantees the existence of Green's function for negatively curved Riemannian surfaces.  It is easy to check that if the series converges for some x ∈M (respectively, p ∈ M ), then it converges for all x ∈M (respectively, p ∈ M ).
In [19] it was proved that the exponent of convergence of a complete negatively curved manifold is equal to the Hausdorff dimension of its conic limit set (see the definition in [23]); it was previously proved in the case of constant curvature in [18] for surfaces of finite area, and in full generality in [11] and [5]. In particular, this fact implies that the set of bounded geodesics in the manifold (geodesics which are contained in a compact set, which depends of each geodesic) has Hausdorff dimension equal to the exponent of convergence (see [5], [11] and [19]). The exponent of convergence also plays an important role in the study of escaping geodesics in negatively curved surfaces (see [12] and [16]  This inequality for the heat kernel allows to obtain an upper bound for the Green's function as follows.
Theorem 3.2. For each n ≥ 2 and r 0 > 0, there exists a positive constant C 1 , which just depends on n and r 0 , with the following property: for every n-dimensional Cartan-Hadamard manifold with sectional curvatures K ≤ −1, the Green's function satisfies for every x, y ∈ M with r := d(x, y) ≥ r 0 .
Using the well known formula g(x, y) = ∞ 0 p(x, y, t) dt and Theorem 3.1, we obtain for every x, y ∈ M with r = d(x, y) ≥ r 0 (making the change s = r 2 /t in the first integral) ≤ c n e −(n−1)r/2 (1 + r + r 2 ) (n−3)/2 (1 + r) The upper bound obtained in Theorem 3.2 for the Green's function in a Cartan-Hadamard manifold allows to obtain a criteria for existence of the Green's function in a negatively curved manifold with arbitrary topology. Since Γ is a discrete group of isometries ofM , for each fixed choice of x, y ∈M , we have that Γ xy := γ ∈ Γ : d(x, γy) < 1 is a finite set. Hence, applying Theorem 3.2, As a consequence of the last theorem, we deduce the following result.    Proof. Theorem 2.3 gives that the harmonic measure of the ideal boundary of M vanishes. Fix a normal domain Ω 0 ⊂ M . For each ε > 0 we can find a normal domain Ω with compact closure, Ω 0 ⊂ Ω and Ω\Ω0 grad u Ω 2 < ε .
Let us define a function f on M as follows: on Ω \ Ω 0 , 0 , on M \ Ω .
The inequality in Theorem 3.7 is sharp as the following examples show: As a consequence of Theorems 3.5 and 3.7, if M is any complete n-dimensional Riemannian manifold without Green's function and with sectional curvatures K = −1, then δ = n − 1.