The principal eigenvalue and maximum principle for second order elliptic operators on Riemannian manifolds
\textit{H. Berestycki}, \textit{L. Nirenberg} and \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 47, No. 1, 47-92 (1994; Zbl 0806.35129)] and \textit{H. Berestycki} and \textit{L. Nirenberg} [Bol. Soc. Bras. Mat., Nova Sér. 22, No. 1, 1-37 (1991; Zbl 0784.35025)] wrote on the maximum principle and principal eigenvalues of second order operators on general domains. The author extends this work to Riemannian manifolds. He shows that the refined maximum principle holds for a second order elliptic operator on a manifold if and only if the principal eigenvalue is positive. The work is motivated by the study of symmetry properties of positive solutions of elliptic equations on symmetric domains.
scientific article; zbMATH DE number 1005656