Mean Curvature Flow and Isoperimetric Inequalities

Mean Curvature Flow and Isoperimetric Inequalities PDF

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.

Isoperimetric Inequalities in Riemannian Manifolds

Isoperimetric Inequalities in Riemannian Manifolds PDF

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.

Isoperimetric Inequality and Area Growth of Surfaces with Bounded Mean Curvature

Isoperimetric Inequality and Area Growth of Surfaces with Bounded Mean Curvature PDF

Author: Dechang Chen

Publisher:

Published: 2014

Total Pages: 60

ISBN-13:

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In this thesis, we give a lower bound on the areas of small geodesic balls in an immersed hypersurface M contained in a Riemannian manifold N. This lower bound depends only on an upper bound for the absolute mean curvature function of M, an upper bound of the absolute sectional curvature of N and a lower bound for the injectivity radius of N. As a consequence, we prove that if M is a noncompact complete surface of bounded absolute mean curvature in Riemannian manifold N with positive injectivity radius and bounded absolute sectional curvature, then the area of geodesic balls of M must grow at least linearly in terms of their radius. In particular, this result implies the classical result of Yau that a complete minimal hypersurface in Rn must have infinite area. We also attain partial results on the conjecture: If M is a compact immersed surface in hyperbolic 3-space H3, and the absolute mean curvature function of M is bounded from above by 1, then Area(M)

Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations

Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations PDF

Author: Giovanni Bellettini

Publisher: Springer

Published: 2014-05-13

Total Pages: 336

ISBN-13: 8876424296

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The aim of the book is to study some aspects of geometric evolutions, such as mean curvature flow and anisotropic mean curvature flow of hypersurfaces. We analyze the origin of such flows and their geometric and variational nature. Some of the most important aspects of mean curvature flow are described, such as the comparison principle and its use in the definition of suitable weak solutions. The anisotropic evolutions, which can be considered as a generalization of mean curvature flow, are studied from the view point of Finsler geometry. Concerning singular perturbations, we discuss the convergence of the Allen–Cahn (or Ginsburg–Landau) type equations to (possibly anisotropic) mean curvature flow before the onset of singularities in the limit problem. We study such kinds of asymptotic problems also in the static case, showing convergence to prescribed curvature-type problems.

Lecture Notes on Mean Curvature Flow

Lecture Notes on Mean Curvature Flow PDF

Author: Carlo Mantegazza

Publisher: Springer Science & Business Media

Published: 2011-07-28

Total Pages: 175

ISBN-13: 3034801459

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This book is an introduction to the subject of mean curvature flow of hypersurfaces with special emphasis on the analysis of singularities. This flow occurs in the description of the evolution of numerous physical models where the energy is given by the area of the interfaces. These notes provide a detailed discussion of the classical parametric approach (mainly developed by R. Hamilton and G. Huisken). They are well suited for a course at PhD/PostDoc level and can be useful for any researcher interested in a solid introduction to the technical issues of the field. All the proofs are carefully written, often simplified, and contain several comments. Moreover, the author revisited and organized a large amount of material scattered around in literature in the last 25 years.

Regularity Theory for Mean Curvature Flow

Regularity Theory for Mean Curvature Flow PDF

Author: Klaus Ecker

Publisher: Springer Science & Business Media

Published: 2004

Total Pages: 192

ISBN-13: 9780817632434

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* Devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flow. * Mean curvature flow and related geometric evolution equations are important tools in mathematics and mathematical physics.

Isoperimetric Inequalities

Isoperimetric Inequalities PDF

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.

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