Local Fatou theorem and the density of energy on manifolds of negative curvature (Q997806)

\(M\) is a complete, simply connected Riemannian manifold, with sectional curvatures bounded between two negative constants. The geometric boundary \(\partial M\), defined by the geodesic rays, agrees with the Martin boundary of \(M\), with respect to the Laplace-Beltrami operator. The boundary carries the family \(\mu = (\mu _ {x})_ {x\in M}\) of equivalent harmonic measures. The density of energy (for some harmonic function \(u\) on \(M\)) is defined as: \[ D_ {c} ^ {0} = -\frac{1}{2} \int _ {\Gamma _ {c}^ {\theta}} \Delta | u|\,dx \] where \(c>0\), \(\theta \in \partial M\), \(\gamma _ {\theta }\) is the geodesic ray from the basepoint \(o\in M\) to \(\theta \) and \[ \Gamma _ {c} ^ {\theta } := \{ x \in M \mid d(x, \gamma _ {\theta } ) < c \} \] The main result asserts that a harmonic function \(u\) on \(M\) is non-tangentially convergent at \(\mu \)-almost all \(\theta \) such that \(D_ {c} ^ {0}\) is finite.