A Laplacian comparison theorem and its applications.
Let \(M\) be a connected complete Riemannian \(n(>1)\)-dimensional manifold, \(p\) be a fixed point in \(M\) and \(h(x)=d(x, p)\) be the geodesic distance from \(p\) to \(x\). Let \(K_M\) and Ric\(_M\) denote the sectional curvature and Ricci curvature of \(M\), respectively. Let \(0<l{\leq}{\infty}\) and \({\gamma}: [0, l){\rightarrow}M\) be a geodesic with \({\gamma}(0)=p, | {\gamma}'(0)| =1\). Under the assumption that \(K_M({\gamma}'(t), X){\leq}k(t)\) with \(X{\in} M_{{\gamma}(t)}\) and \(X{\perp}{\gamma}'(t)\), \textit{A. Kasue} [Jap. J. Math., New Ser. 8, 309--341 (1982; Zbl 0518.53048)] proved the following Laplace comparison theorem: Let \(f\) be a solution of \(f''+k(t)f=0,\; f(0)=0,\; f'(0)=1\) then \[ {\bigtriangleup}h({\gamma}(t)){\geq}(n-1)f'/f. \] The author of the present paper assumes further that Ric\(_M({\gamma}'(t), {\gamma}'(t)){\leq}r(t){\leq}(n-1)k(t)\) and generalizes the Laplace comparison theorem as follows: Let \(f_1\) be a solution of \(f_1''+\{r(t)-(n-2)k(t)\}f_1=0,\; f_1(0)=0,\; f_1(0)=1,\) then \[ f_1(t){\geq}f(t),\; f_1'(t)/f_1(t){\geq}f'(t)/f(t),\; {\bigtriangleup}(h({\gamma}(t)){\geq}(n-2)f'(t)/f(t)+f_1'(t)/f_1'(t). \] Using this extended Laplace comparison theorem, the author obtains a volume comparison theorem and a criterion for the hyperbolicity of Riemannian manifolds, i.e., shows the existence of Green's function for these manifolds.
scientific article; zbMATH DE number 1782978
scientific article