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This book presents very recent results involving an extensive use of analytical tools in the study of geometrical and topological properties of complete Riemannian manifolds. It analyzes in detail an extension of the Bochner technique to the non compact setting, yielding conditions which ensure that solutions of geometrically significant differential equations either are trivial (vanishing results) or give rise to finite dimensional vector spaces (finiteness results). The book develops a range of methods from spectral theory and qualitative properties of solutions of PDEs to comparison theorems in Riemannian geometry and potential theory. All needed tools are described in detail, often with an original approach. Some of the applications presented concern the topology at infinity of submanifolds, Lp cohomology, metric rigidity of manifolds with positive spectrum, and structure theorems for Kähler manifolds. The book is essentially self-contained and supplies in an original presentation the necessary background material not easily available in book form.

Riemannian manifolds. --- Bochner technique. --- Geometry, Riemannian. --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Riemann geometry --- Riemannian geometry --- Generalized spaces --- Geometry, Non-Euclidean --- Semi-Riemannian geometry --- Geometry, Differential --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Manifolds (Mathematics) --- Global differential geometry. --- Global analysis. --- Global analysis (Mathematics). --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Analysis. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential geometry. --- Manifolds (Mathematics). --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis --- Topology --- Differential geometry

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Partial differential equations --- Differential equations, Parabolic. --- Heat equation. --- Diffusion processes. --- Differential equations, Parabolic --- Diffusion processes --- Heat equation --- Diffusion equation --- Heat flow equation --- Markov processes --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial

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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.

Geometry --- Mathematics --- Physical Sciences & Mathematics --- Maximum principles (Mathematics) --- Geometric analysis. --- Geometric analysis PDEs (Geometric partial differential equations) --- Mathematical analysis --- Differential equations, Partial --- Numerical solutions --- Global analysis. --- Differential equations, partial. --- Geometry. --- Global Analysis and Analysis on Manifolds. --- Partial Differential Equations. --- Euclid's Elements --- Partial differential equations --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Partial differential equations. --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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Differential geometry. Global analysis --- differentiaal geometrie

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The aim of this monograph is to present a self-contained introduction to some geometric and analytic aspects of the Yamabe problem. The book also describes a wide range of methods and techniques that can be successfully applied to nonlinear differential equations in particularly challenging situations. Such situations occur where the lack of compactness, symmetry and homogeneity prevents the use of more standard tools typically used in compact situations or for the Euclidean setting. The work is written in an easy style that makes it accessible even to non-specialists. After a self-contained treatment of the geometric tools used in the book, readers are introduced to the main subject by means of a concise but clear study of some aspects of the Yamabe problem on compact manifolds. This study provides the motivation and geometrical feeling for the subsequent part of the work. In the main body of the book, it is shown how the geometry and the analysis of nonlinear partial differential equations blend together to give up-to-date results on existence, nonexistence, uniqueness and a priori estimates for solutions of general Yamabe-type equations and inequalities on complete, non-compact Riemannian manifolds.

Differential geometry. Global analysis --- differentiaal geometrie