Spectrum, harmonic functions, and hyperbolic metric spaces
type: Item
instance of: scholarly article
title: Spectrum, harmonic functions, and hyperbolic metric spaces
zbMATH Open document ID: 0840.31008
DOI: 10.1007/BF02808210
author: Yuri Kifer
published in: Israel Journal of Mathematics
publication date: 1996-07-04T00:00:00Z
review text: For Riemannian manifolds \(M\) the author generalizes former results concerning the spectrum of the Laplace-Beltrami operator \(\Delta\) and the Martin boundary. \(M\) is assumed to be a complete, simply connected \(C^3\) Riemannian manifold without focal points. Moreover, it is assumed that \(M\) is also a hyperbolic metric space in the sense of Gromov. This condition relaxes conditions of former articles, where usually some bounds for the sectional curvatures were assumed. The main result is the heat kernel estimate \[ p(t,x,y) \leq c^{-1} t^{- n/2} \biggl( \exp (- ct) \wedge \exp \bigl( - cd (x,y) \bigr) \biggr), \] where \(c > 0\) is a constant and \(d\) is the distance function. This implies for the top \(\lambda\) of the \(L^2\)-spectrum of \(\Delta\) that \(\lambda \leq - c\), which was known if the sectional curvatures are bounded from below and above by negative constants. Actually the theorem is proven under a condition which for \(\dim M > 2\) is weaker than hyperbolicity. For \( \dim M = 2\) both conditions turn out to be equivalent to some growth condition for the volume of balls. The sphere \(S (\infty)\) of classes of asymptotic geodesics defines as a boundary at infinity a natural compactification of \(M\). The author proves the following results: The Dirichlet problem at \(\infty\) has a unique solution, the corresponding harmonic measures on \(S (\infty)\) have no atoms and positive Hausdorff dimension. The set of minimal positive harmonic functions coincides with the Martin boundary of \(M\) and is homeomorphic to \(S (\infty)\). These results can also be proved using methods of Ancona as corollaries to the first theorem.
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 755554
zbMATH Keywords: hyperbolic metric space
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: spectrum
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Martin boundary
zbMATH Keywords: heat kernel estimate
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: harmonic measures
reviewed by: Walter Hoh
MaRDI profile type: Publication
cites work: Negatively curved manifolds, elliptic operators, and the Martin boundary
cites work: Q5751700
cites work: \(C^ \alpha\)-compactness for manifolds with Ricci curvature and injectivity radius bounded below
cites work: Positive harmonic functions on complete manifolds of negative curvature
cites work: On the Dirichlet Problem at Infinity for Manifolds of Nonpositive Curvature
cites work: The fundamental group and the spectrum of the Laplacian
cites work: Q5341449
cites work: Harmonic Measures on Covers of Compact Surfaces of Nonpositive Curvature
cites work: Q4010477
cites work: Q3344779
cites work: Some isoperimetric inequalities and eigenvalue estimates
cites work: Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov. (Geometry and group theory. The hyperbolic groups of Gromov)
cites work: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Geodesic Flow in Certain Manifolds Without Conjugate Points
cites work: When is a geodesic flow of Anosov type, I
cites work: When is a geodesic flow of Anosov type, II
cites work: Another note on the Borel-Cantelli lemma and the strong law, with the Poisson approximation as a by-product
cites work: Manifolds without focal points
cites work: The cone topology on a manifold without focal points
cites work: Q3993500
cites work: Q3772406
cites work: On the Variety of Manifolds without Conjugate Points
cites work: Q5538378
cites work: Q3959169
cites work: Four applications of conformal equivalence to geometry and dynamics
cites work: Q3751848
cites work: Hausdorff Dimension of Harmonic Measures on Negatively Curved Manifolds
cites work: Q4039796
cites work: Q4727002
cites work: On the parabolic kernel of the Schrödinger operator
cites work: Morse Theory. (AM-51)
cites work: An upper bound to the spectrum of \(\Delta\) on a manifold of negative curvature
description: scientific article; zbMATH DE number 755554