## 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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### References:

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