Isoperimetric Inequalities in Riemannian Manifolds

Manuel Ritoré

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English

Birkhauser Verlag AG
07 October 2023

Calculus & mathematical analysis; Differential & Riemannian geometry; Optimisation

Series: Progress in Mathematics

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.

By: Manuel Ritoré
Imprint: Birkhauser Verlag AG
Country of Publication: Switzerland
Edition: 2023 ed.
Volume: 348
Dimensions: Height: 235mm, Width: 155mm,
Weight: 881g
ISBN: 9783031379000
ISBN 10: 3031379004
Series: Progress in Mathematics
Pages: 460
Publication Date: 07 October 2023
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

- 1. Introduction. - 2. Isoperimetric Inequalities in Surfaces. - 3. The Isoperimetric Profile of Compact Manifolds. - 4. The Isoperimetric Profile of Non-compact Manifolds. - 5. Symmetrization and Classical Results. - 6. Space Forms. - 7. The Isoperimetric Profile for Small and Large Volumes. - 8. Isoperimetric Comparison for Sectional Curvature. - 9. Relative Isoperimetric Inequalities.

Manuel Ritoré is professor of Mathematics at the University of Granada since 2007. His earlier research focused on geometric inequalities in Riemannian manifolds, specially on those of isoperimetric type. In this field he has obtained some results such as a classification of isoperimetric sets in the 3-dimensional real projective space; a classification of 3-dimensional double bubbles; existence of solutions of the Allen-Cahn equation near non-degenerate minimal surfaces; an alternative proof of the isoperimetric conjecture for 3-dimensional Cartan-Hadamard manifolds; optimal isoperimetric inequalities outside convex sets in the Euclidean space; and a characterization of isoperimetric regions of large volume in Riemannian cylinders, among others. Recently, he has become interested on geometric variational problems in spaces with less regularity, such as sub-Riemannian manifolds or more general metric measure spaces, where he has obtained a classification of isoperimetric sets inthe first Heisenberg group under regularity assumptions, and Brunn-Minkowski inequalities for metric measure spaces, among others.

Reviews for Isoperimetric Inequalities in Riemannian Manifolds

“The book presents well-written proofs, with detailed explanations, ensuring that they are both complete and accessible, and the clear writing style makes it a pleasure to read. All these qualities make it a must-read for anyone interested in isoperimetric inequalities.” (Giorgio Saracco, zbMATH 1557.53002, 2025)