Local Fatou theorem and the density of energy on manifolds of negative curvature
type: Item
instance of: scholarly article
title: Local Fatou theorem and the density of energy on manifolds of negative curvature
zbMATH Open document ID: 1131.31004
DOI: 10.4171/RMI/484
author: Frédéric Mouton
published in: Revista Matemática Iberoamericana
publication date: 2007-08-07T00:00:00Z
full work available at URL: https://eudml.org/doc/43581
review text: \(M\) 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.
reviewed by: Liliana Popa
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31C35
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 5177703
zbMATH Keywords: harmonic functions
zbMATH Keywords: Fatou type theorems
zbMATH Keywords: negative curvature
zbMATH Keywords: Brownian motion
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5177703