Spectrum, harmonic functions, and hyperbolic metric spaces
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.
scientific article; zbMATH DE number 755554
scientific article