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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.
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.
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.
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.
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.
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.
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