The aim of the paper is to introduce the reader to various forms of the maximum principle, starting from its classical formulation up to generalizations of the Omori-Yau maximum principle at infinity recently obtained by the authors. Applications are given to a number of geometrical problems in the setting of complete Riemannian manifolds, under assumptions either on the curvature or on the volume growth of geodesic balls.

Readership

Graduate students and research mathematicians interested in analysis and Riemannian geometry.

Table of Contents

• Preliminaries and some geometric motivations
• Further typical applications of Yau's technique
• Stochastic completeness and the weak maximum principle
• The weak maximum principle for the \(\varphi\)-Laplacian
• \(\varphi\)-parabolicity and some further remarks
• Curvature and the maximum principle for the \(\varphi\)-Laplacian
• Bibliography