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Local Fatou theorem and the density of energy on manifolds of negative curvature. (English) Zbl 1131.31004

\(M\) 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.

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
31C35 Martin boundary theory
58J65 Diffusion processes and stochastic analysis on manifolds
60J45 Probabilistic potential theory
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