Isoperimetric inequalities. Differential geometric and analytic perspectives (Q2744685)

``The isoperimetric problem has been a source of mathematical ideas and techniques since its formulation in classical antiquity, and it is still alive and well in its ability to both capture and nourish the mathematical imagination.'' A new book by \textit{Isaac Chavel} [who is the author of the well-known ``Eigenvalues in Riemannian Geometry'' (1984; Zbl 0551.53001) and ``Riemannian Geometry: A Modern Introduction'' (1995; Zbl 0819.53001)] treats the classical isoperimetric inequalities in Euclidean space and in noncompact Riemannian manifolds. In Euclidean space the emphasis is on quantitative precision for very general domains, and in Riemannian manifolds the emphasis is on qualitative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques and applications to upper bounds of large time heat diffusion in Riemannian manifolds. NEWLINENEWLINENEWLINEThe more detailed description of the content of the book is as follows. Chapter I starts with posing the isoperimetric problem in Euclidean space and gives some elementary arguments toward its solution in the Euclidean plane. Chapter II starts with uniqueness theory, under the assumption that the boundary of the solution domain is \(C^2\). For domains with \(C^1\) boundary the problem of existence is considered. Chapter III is the heart of the first half of the book. It expands the isoperimetric problem for all compacta in Euclidean space. In this setting, using Blaschke selection theorem and Steiner symmetrization, one shows that the closed disk constitutes a solution to the isoperimetric problem. Chapter IV introduces the Hausdorff measure for subsets of Euclidean space, the area formula is proved. NEWLINENEWLINENEWLINEThe second half of the book begins with Chapter V introducing a new view of isoperimetric inequalities, namely, rough inequalities in a Riemannian manifold. The goal of succeeding chapters V--VIII is to show how these geometric isoperimetric inequalities influence the qualitative rate of decay, with respect to time, of heat diffusion in Riemannian manifolds. Chapter V summarizes the basic notions and results concerning isoperimetric inequalities in Riemannian manifolds, and Chapter VI gives their implications for analytic Sobolev inequalities on Riemannian manifolds. Chapter VII introduces the Laplacian and the heat operator on Riemannian manifolds and is devoted to setting the stage for the ``main problem'' in large time heat diffusion; its formulation and solution are presented in Chapter VIII. The book ends with an introduction to the new arguments of A. A. Grigor'yan. NEWLINENEWLINENEWLINEThe book provides the right balance between merely summarizing background material and developing preparatory arguments in the text. Although it is required occasionally some use of the material which will be introduced only lately in forthcoming chapters.NEWLINENEWLINENEWLINEThe reader gets the introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities expressing its beauty and inspiration, a subject that has beginnings in classical antiquity and that continues to inspire fresh ideas in geometry and analysis to this very day -- and beyond.