Harmonic functions on manifolds of negative curvature (Q1093045)

Let \(M\) be a complete simply-connected \(n\)-dimensional manifold having sectional curvature at most \(-k^2\), where \(k^2>0\). Then: (i) the Green function \(E(x)\) for \(M\) with the pole at a point \(0\in M\) satisfies \(E(x)\leq (k^{n-1}/\omega_n)\int^\infty_r(1/sh^{n-1}kt)\,dt\), (ii) the mean value theorem is proved for harmonic functions on \(M\). If such a function belongs to \(L^1(M)\), then it is equal to zero identically.