09-15-2026, 11:13 PM
“Global Analysis: An Introduction to Nonlinear Analysis and Its Variational Methods on Riemannian Manifolds” by Carlos Daniel Velázquez-Mendoza, María de los Ángeles Sandoval-Romero, and Romulo Diaz Carlos is a comprehensive monograph introducing global analysis through the interaction of differential geometry, functional analysis, partial differential equations, and variational methods.
It begins with smooth and Riemannian geometry and develops the analytical machinery needed for modern geometric PDEs, including Sobolev spaces, distributions, interpolation, fractional regularity, differential and pseudodifferential operators, elliptic theory, heat methods, bounded geometry, and trace theorems. The emphasis is on explaining how local Euclidean estimates can be converted into intrinsic statements on manifolds, while carefully identifying the geometric assumptions required in compact, noncompact, and boundary settings.
The later sections move into major themes of modern global analysis: Fredholm and index theory, culminating in the Atiyah–Singer index theorem, geometric evolution equations such as Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and nonlinear variational techniques including the direct method, Palais–Smale condition, deformation arguments, mountain-pass theorem, and Nehari method.
Rather than presenting these subjects as isolated theories, the monograph aims to show how geometry, analysis, and variational principles fit together to study nonlinear equations and geometric phenomena. It is intended primarily for advanced undergraduate and graduate students and for readers entering global analysis from neighboring areas such as differential geometry or PDEs.
BOOK
It begins with smooth and Riemannian geometry and develops the analytical machinery needed for modern geometric PDEs, including Sobolev spaces, distributions, interpolation, fractional regularity, differential and pseudodifferential operators, elliptic theory, heat methods, bounded geometry, and trace theorems. The emphasis is on explaining how local Euclidean estimates can be converted into intrinsic statements on manifolds, while carefully identifying the geometric assumptions required in compact, noncompact, and boundary settings.
The later sections move into major themes of modern global analysis: Fredholm and index theory, culminating in the Atiyah–Singer index theorem, geometric evolution equations such as Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and nonlinear variational techniques including the direct method, Palais–Smale condition, deformation arguments, mountain-pass theorem, and Nehari method.
Rather than presenting these subjects as isolated theories, the monograph aims to show how geometry, analysis, and variational principles fit together to study nonlinear equations and geometric phenomena. It is intended primarily for advanced undergraduate and graduate students and for readers entering global analysis from neighboring areas such as differential geometry or PDEs.
BOOK
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