Brakke's Mean Curvature Flow

Brakke's Mean Curvature Flow
Title Brakke's Mean Curvature Flow PDF eBook
Author Yoshihiro Tonegawa
Publisher Springer
Pages 108
Release 2019-04-09
Genre Mathematics
ISBN 9811370753

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This book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 ≤ k in

Regularity Theory for Mean Curvature Flow

Regularity Theory for Mean Curvature Flow
Title Regularity Theory for Mean Curvature Flow PDF eBook
Author Klaus Ecker
Publisher Springer Science & Business Media
Pages 173
Release 2012-12-06
Genre Mathematics
ISBN 0817682104

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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.

Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations

Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations
Title Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations PDF eBook
Author Giovanni Bellettini
Publisher Springer
Pages 336
Release 2014-05-13
Genre Mathematics
ISBN 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.

Mean Curvature Flow and Isoperimetric Inequalities

Mean Curvature Flow and Isoperimetric Inequalities
Title Mean Curvature Flow and Isoperimetric Inequalities PDF eBook
Author Manuel Ritoré
Publisher Springer Science & Business Media
Pages 113
Release 2010-01-01
Genre Mathematics
ISBN 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.

Elliptic Regularization and Partial Regularity for Motion by Mean Curvature

Elliptic Regularization and Partial Regularity for Motion by Mean Curvature
Title Elliptic Regularization and Partial Regularity for Motion by Mean Curvature PDF eBook
Author Tom Ilmanen
Publisher American Mathematical Soc.
Pages 106
Release 1994
Genre Mathematics
ISBN 0821825828

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We study Brakke's motion of varifolds by mean curvature in the special case that the initial surface is an integral cycle, giving a new existence proof by mean of elliptic regularization. Under a uniqueness hypothesis, we obtain a weakly continuous family of currents solving Brakke's motion. These currents remain within the corresponding level-set motion by mean curvature, as defined by Evans-Spruck and Chen-Giga-Goto. Now let [italic capital]T0 be the reduced boundary of a bounded set of finite perimeter in [italic capital]R[superscript italic]n. If the level-set motion of the support of [italic capital]T0 does not develop positive Lebesgue measure, then there corresponds a unique integral [italic]n-current [italic capital]T, [partial derivative/boundary/degree of a polynomial symbol][italic capital]T = [italic capital]T0, whose time-slices form a unit density Brakke motion. Using Brakke's regularity theorem, spt [italic capital]T is smooth [script capital]H[superscript italic]n-almost everywhere. In consequence, almost every level-set of the level-set flow is smooth [script capital]H[superscript italic]n-almost everywhere in space-time.

Lecture Notes on Mean Curvature Flow

Lecture Notes on Mean Curvature Flow
Title Lecture Notes on Mean Curvature Flow PDF eBook
Author Carlo Mantegazza
Publisher Springer Science & Business Media
Pages 175
Release 2011-07-28
Genre Mathematics
ISBN 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.

Space – Time – Matter

Space – Time – Matter
Title Space – Time – Matter PDF eBook
Author Jochen Brüning
Publisher Walter de Gruyter GmbH & Co KG
Pages 590
Release 2018-04-09
Genre Mathematics
ISBN 3110451530

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This monograph describes some of the most interesting results obtained by the mathematicians and physicists collaborating in the CRC 647 "Space – Time – Matter", in the years 2005 - 2016. The work presented concerns the mathematical and physical foundations of string and quantum field theory as well as cosmology. Important topics are the spaces and metrics modelling the geometry of matter, and the evolution of these geometries. The partial differential equations governing such structures and their singularities, special solutions and stability properties are discussed in detail. Contents Introduction Algebraic K-theory, assembly maps, controlled algebra, and trace methods Lorentzian manifolds with special holonomy – Constructions and global properties Contributions to the spectral geometry of locally homogeneous spaces On conformally covariant differential operators and spectral theory of the holographic Laplacian Moduli and deformations Vector bundles in algebraic geometry and mathematical physics Dyson–Schwinger equations: Fix-point equations for quantum fields Hidden structure in the form factors ofN = 4 SYM On regulating the AdS superstring Constraints on CFT observables from the bootstrap program Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities Yangian symmetry in maximally supersymmetric Yang-Mills theory Wave and Dirac equations on manifolds Geometric analysis on singular spaces Singularities and long-time behavior in nonlinear evolution equations and general relativity