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Yamabe-type Equations on Complete, Noncompact Manifolds

2012-07-30 Mathematics
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Release : 2012-07-30
Genre : Mathematics
Kind : eBook
Book Rating : 761/5 ( reviews)

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Book Synopsis Yamabe-type Equations on Complete, Noncompact Manifolds by : Paolo Mastrolia

Download or read book Yamabe-type Equations on Complete, Noncompact Manifolds written by Paolo Mastrolia. This book was released on 2012-07-30. Available in PDF, EPUB and Kindle. Book excerpt: 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.

A Perspective on Canonical Riemannian Metrics

2020-10-23 Mathematics
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Release : 2020-10-23
Genre : Mathematics
Kind : eBook
Book Rating : 858/5 ( reviews)

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Book Synopsis A Perspective on Canonical Riemannian Metrics by : Giovanni Catino

Download or read book A Perspective on Canonical Riemannian Metrics written by Giovanni Catino. This book was released on 2020-10-23. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on a selection of special topics, with emphasis on past and present research of the authors on “canonical” Riemannian metrics on smooth manifolds. On the backdrop of the fundamental contributions given by many experts in the field, the volume offers a self-contained view of the wide class of “Curvature Conditions” and “Critical Metrics” of suitable Riemannian functionals. The authors describe the classical examples and the relevant generalizations. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Maximum Principles and Geometric Applications

2016-02-13 Mathematics
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Release : 2016-02-13
Genre : Mathematics
Kind : eBook
Book Rating : 373/5 ( reviews)

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Book Synopsis Maximum Principles and Geometric Applications by : Luis J. Alías

Download or read book Maximum Principles and Geometric Applications written by Luis J. Alías. This book was released on 2016-02-13. Available in PDF, EPUB and Kindle. Book excerpt: 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.

On Some Aspects of Oscillation Theory and Geometry

2013-08-23 Mathematics
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Release : 2013-08-23
Genre : Mathematics
Kind : eBook
Book Rating : 998/5 ( reviews)

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Book Synopsis On Some Aspects of Oscillation Theory and Geometry by : Bruno Bianchini

Download or read book On Some Aspects of Oscillation Theory and Geometry written by Bruno Bianchini. This book was released on 2013-08-23. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation the authors prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep their investigation basically self-contained, the authors also collect some, more or less known, material which often appears in the literature in various forms and for which they give, in some instances, new proofs according to their specific point of view.

Geometric Analysis of Quasilinear Inequalities on Complete Manifolds

2021-01-18 Mathematics
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Release : 2021-01-18
Genre : Mathematics
Kind : eBook
Book Rating : 047/5 ( reviews)

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Book Synopsis Geometric Analysis of Quasilinear Inequalities on Complete Manifolds by : Bruno Bianchini

Download or read book Geometric Analysis of Quasilinear Inequalities on Complete Manifolds written by Bruno Bianchini. This book was released on 2021-01-18. Available in PDF, EPUB and Kindle. Book excerpt: This book demonstrates the influence of geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, the authors study classes of coercive elliptic differential inequalities on domains of a manifold M with very general nonlinearities depending on the variable x, on the solution u and on its gradient. The book highlights the mean curvature operator and its variants, and investigates the validity of strong maximum principles, compact support principles and Liouville type theorems. In particular, it identifies sharp thresholds involving curvatures or volume growth of geodesic balls in M to guarantee the above properties under appropriate Keller-Osserman type conditions, which are investigated in detail throughout the book, and discusses the geometric reasons behind the existence of such thresholds. Further, the book also provides a unified review of recent results in the literature, and creates a bridge with geometry by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau’s Hessian and Laplacian principles and subsequent improvements.

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