Isoperimetric Inequalities in Riemannian Manifolds
Isoperimetric Inequalities in Riemannian Manifolds

Author: Manuel Ritoré

Publisher: Springer Nature

Published: 2023-10-06

Total Pages: 470

ISBN-13: 3031379012

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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.

Asymptotic Theory of Finite Dimensional Normed Spaces
Asymptotic Theory of Finite Dimensional Normed Spaces

Author: Vitali D. Milman

Publisher: Springer

Published: 2009-02-27

Total Pages: 166

ISBN-13: 3540388222

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This book deals with the geometrical structure of finite dimensional normed spaces, as the dimension grows to infinity. This is a part of what came to be known as the Local Theory of Banach Spaces (this name was derived from the fact that in its first stages, this theory dealt mainly with relating the structure of infinite dimensional Banach spaces to the structure of their lattice of finite dimensional subspaces). Our purpose in this book is to introduce the reader to some of the results, problems, and mainly methods developed in the Local Theory, in the last few years. This by no means is a complete survey of this wide area. Some of the main topics we do not discuss here are mentioned in the Notes and Remarks section. Several books appeared recently or are going to appear shortly, which cover much of the material not covered in this book. Among these are Pisier's [Pis6] where factorization theorems related to Grothendieck's theorem are extensively discussed, and Tomczak-Jaegermann's [T-Jl] where operator ideals and distances between finite dimensional normed spaces are studied in detail. Another related book is Pietch's [Pie].

Isoperimetric Inequalities
Isoperimetric Inequalities

Author: Isaac Chavel

Publisher: Cambridge University Press

Published: 2001-07-23

Total Pages: 292

ISBN-13: 9780521802673

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This advanced introduction emphasizes the variety of ideas, techniques, and applications of the subject.

Mean Curvature Flow and Isoperimetric Inequalities
Mean Curvature Flow and Isoperimetric Inequalities

Author: Manuel Ritoré

Publisher: Springer Science & Business Media

Published: 2010-01-01

Total Pages: 113

ISBN-13: 3034602138

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Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.

An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem
An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

Author: Luca Capogna

Publisher: Springer Science & Business Media

Published: 2007-08-08

Total Pages: 235

ISBN-13: 3764381337

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This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

Riemannian Geometry
Riemannian Geometry

Author: Isaac Chavel

Publisher: Cambridge University Press

Published: 1995-01-27

Total Pages: 402

ISBN-13: 9780521485784

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This book provides an introduction to Riemannian geometry, the geometry of curved spaces. Its main theme is the effect of the curvature of these spaces on the usual notions of geometry, angles, lengths, areas, and volumes, and those new notions and ideas motivated by curvature itself. Isoperimetric inequalities--the interplay of curvature with volume of sets and the areas of their boundaries--is reviewed along with other specialized classical topics. A number of completely new themes are created by curvature: they include local versus global geometric properties, that is, the interaction of microscopic behavior of the geometry with the macroscopic structure of the space. Also featured is an ambitious "Notes and Exercises" section for each chapter that will develop and enrich the reader's appetite and appreciation for the subject.

Eigenvalues in Riemannian Geometry
Eigenvalues in Riemannian Geometry

Author: Isaac Chavel

Publisher: Academic Press

Published: 1984-11-07

Total Pages: 379

ISBN-13: 0080874347

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The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery.

Geometric Inequalities
Geometric Inequalities

Author: Yurii D. Burago

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 346

ISBN-13: 3662074419

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A 1988 classic, covering Two-dimensional Surfaces; Domains on the Plane and on Surfaces; Brunn-Minkowski Inequality and Classical Isoperimetric Inequality; Isoperimetric Inequalities for Various Definitions of Area; and Inequalities Involving Mean Curvature.

Needle Decompositions in Riemannian Geometry
Needle Decompositions in Riemannian Geometry

Author: Bo’az Klartag

Publisher: American Mathematical Soc.

Published: 2017-09-25

Total Pages: 90

ISBN-13: 1470425424

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The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in the author's analysis.

New Trends on Analysis and Geometry in Metric Spaces
New Trends on Analysis and Geometry in Metric Spaces

Author: Fabrice Baudoin

Publisher: Springer Nature

Published: 2022-02-04

Total Pages: 312

ISBN-13: 3030841413

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This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.