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

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Local Fatou theorem and the density of energy on manifolds of negative curvature y Frederic Mouton Abstract Let u be a harmonic function on a complete simply connected manifold M whose sectional curvatures are bounded between two negative constants. It is proved here a pointwise criterion of non- tangential convergence for points of the geometric boundary: the niteness of the density of energy, which is the geometric analogue of the density of the area integral in the Euclidean half-space. Introduction If the study of non-tangential convergence of harmonic functions began in 1906 with the well-known theorem of Fatou (see [11]), it became clear in the 1970s (see for example [14]) that spaces of negative curvature provide a nat- ural \geometric setting for this study: as can be seen in section 1, several notions have simpler or more natural expressions in this geometric setting. From this point of view, we proved some years ago two pointwise criteria of non-tangential convergence | non-tangential boundedness and niteness of the non-tangential energy | on Riemannian manifolds of pinched negative curvature by the use of Brownian motion (see [16]). We refer the reader to this article for the historical details and references on the study of non- tangential convergence.

tangential convergence

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diusion associated

associated ?-algebra

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brownian motion

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