Maximum Principles and Geometric Applications

Authors: Alías, Luis J., Mastrolia, Paolo, Rigoli, Marco
Provides a self-contained approach to the study of geometric and analytic aspects of maximum principles, making it a perfect companion to other books on the subject
Presents the essential analytic tools and the geometric foundations needed to understand maximum principles and their geometric applications
Includes a wide range of applications of maximum principles to different geometric problems, including some topics that are rare in current literature such as Ricci solitons
Relevant to other areas of mathematics, namely, partial differential equations on manifolds, calculus of variations, and probabilistic potential theory

This monograph presents an introduction to some geometric and analytic aspects of the maximum principle. In doing so, it analyses with great detail the mathematical tools and geometric foundations needed to develop the various new forms that are presented in the first chapters of the book. In particular, a generalization of the Omori-Yau maximum principle to a wide class of differential operators is given, as well as a corresponding weak maximum principle and its equivalent open form and parabolicity as a special stronger formulation of the latter.
In the second part, the attention focuses on a wide range of applications, mainly to geometric problems, but also on some analytic (especially PDEs) questions including: the geometry of submanifolds, hypersurfaces in Riemannian and Lorentzian targets, Ricci solitons, Liouville theorems, uniqueness of solutions of Lichnerowicz-type PDEs and so on.
Maximum Principles and Geometric Applications is written in an easy style making it accessible to beginners. The reader is guided with a detailed presentation of some topics of Riemannian geometry that are usually not covered in textbooks. Furthermore, many of the results and even proofs of known results are new and lead to the frontiers of a contemporary and active field of research.

Topics
Global Analysis and Analysis on Manifolds
Partial Differential Equations
Geometry

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