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2023 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Isoperimetric Inequalities in Riemannian Manifolds

verfasst von: Manuel Ritoré

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This work gives a coherent introduction to isoperimetric inequalities in Riemannian manifolds, featuring many of the results obtained during the last 25 years and discussing different techniques in the area.
Written in a clear and appealing style, the book includes sufficient introductory material, making it also accessible to graduate students. It will be of interest to researchers working on geometric inequalities either from a geometric or analytic point of view, but also to those interested in applying the described techniques to their field.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
In this chapter, we collect some basic material that will be widely used throughout this work. Some knowledge of well-known results in Riemannian geometry will be assumed.
Manuel Ritoré
Chapter 2. Isoperimetric Inequalities in Surfaces
Abstract
In this chapter, we study isoperimetric inequalities in Riemannian surfaces. Firstly, we present Blaschke’s variational proof of the Gauss-Bonnet theorem, a fundamental tool in the two-dimensional isoperimetric theory. Then we recall Hurwitz’s simple proof of the isoperimetric inequality in the plane using Wirtinger’s inequality and, afterward, Weil’s proof of the validity of the planar isoperimetric inequality in Cartan-Hadamard planes (i.e., complete simply connected surfaces with non-positive sectional Gauss curvature).
Manuel Ritoré
Chapter 3. The Isoperimetric Profile of Compact Manifolds
Abstract
In this chapter, the basic properties of the isoperimetric profile \(I_M\) of a compact Riemannian manifold M, and of their isoperimetric sets, are described. We recall that the isoperimetric profile is the function that assigns to any volume the infimum of the perimeter of sets of this volume and should be thought of as the best possible isoperimetric inequality in M. Isoperimetric regions are the ones with the smallest possible perimeter for a given volume.
Manuel Ritoré
Chapter 4. The Isoperimetric Profile of Non-compact Manifolds
Abstract
In this chapter, we consider complete non-compact Riemannian manifolds. From the isoperimetric point of view, several differences with the compact case immediately arise, like the lack of continuity of the isoperimetric profile and the phenomenon of non-existence of isoperimetric regions.
Manuel Ritoré
Chapter 5. Symmetrization and Classical Results
Abstract
In the theory of isoperimetric inequalities, a symmetrization of a setE is a procedure to obtain another set \(E^*\) with the same volume as E, no larger perimeter, and additional symmetries. Symmetrization is a very powerful method to reduce the number of isoperimetric candidates in a given space and generally requires that the ambient manifold itself possesses symmetries.
Manuel Ritoré
Chapter 6. Space Forms
Abstract
In this chapter, isoperimetric inequalities in space forms, complete Riemannian manifolds with constant sectional curvatures, are considered. We have already seen in Sect. 4.​5 that isoperimetric sets in simply connected space forms are geodesic balls. When considering manifolds with non-trivial topology, we should expect existence of isoperimetric sets of different topological types. This often implies the existence of several explicit isoperimetric inequalities, one for each topological type of isoperimetric solution. A way of dealing with this phenomenon is through a classification of stable hypersurfaces, second-order minima of the perimeter under volume-preserving variations.
Manuel Ritoré
Chapter 7. The Isoperimetric Profile for Small and Large Volumes
Abstract
In this chapter, a few results on the isoperimetric profile for small and large volumes are presented. The behavior of the profile for small volumes in compact manifolds was treated in an elementary way in Sect. 3.​3 in Chap. 3, where a result by Bérard and Meyer [52] was proven, namely, that the isoperimetric profile for small volumes in compact manifolds is asymptotic to the one of Euclidean space.
Manuel Ritoré
Chapter 8. Isoperimetric Comparison for Sectional Curvature
Abstract
We consider in this chapter the validity of the Euclidean isoperimetric inequality in a Cartan–Hadamard manifold, a complete, simply connected manifold with non-negative sectional curvature \(K_{\sec }\).
Manuel Ritoré
Chapter 9. Relative Isoperimetric Inequalities
Abstract
In this chapter, isoperimetric inequalities on domains with smooth boundary in complete Riemannian manifolds are considered. The perimeter here is the one relative to the domain \(\Omega \). If a smooth interface S separates \(\Omega \) into two sets, the relative perimeter of each one of these sets is the area of the interface. This means that there are no contributions to the relative perimeter coming from pieces in \(\partial \Omega \). While many of the techniques are adapted from the boundaryless case, some subtle differences appear. The geometry of the boundary, in particular its second fundamental form, will play a determinant role.
Manuel Ritoré
Backmatter
Metadaten
Titel
Isoperimetric Inequalities in Riemannian Manifolds
verfasst von
Manuel Ritoré
Copyright-Jahr
2023
Electronic ISBN
978-3-031-37901-7
Print ISBN
978-3-031-37900-0
DOI
https://doi.org/10.1007/978-3-031-37901-7

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