Regularity Theory for Mean Curvature Flow
Author: Klaus Ecker
Publisher:
Published: 2011-04-26
Total Pages: 184
ISBN-13: 9780817682118
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Regularity Theory for Mean Curvature Flow
Author: Klaus Ecker
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 173
ISBN-13: 0817682104
DOWNLOAD EBOOK* 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.
Regularity Theory for Mean Curvature Flow
Author: K. Ecker
Publisher:
Published: 2004
Total Pages:
ISBN-13: 9783764337810
DOWNLOAD EBOOKBrakke's Mean Curvature Flow
Author: Yoshihiro Tonegawa
Publisher: Springer
Published: 2019-04-09
Total Pages: 108
ISBN-13: 9811370753
DOWNLOAD EBOOKThis 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
Elliptic Regularization and Partial Regularity for Motion by Mean Curvature
Author: Tom Ilmanen
Publisher: American Mathematical Soc.
Published: 1994
Total Pages: 106
ISBN-13: 0821825828
DOWNLOAD EBOOKWe 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
Author: Carlo Mantegazza
Publisher: Springer Science & Business Media
Published: 2011-07-28
Total Pages: 175
ISBN-13: 3034801459
DOWNLOAD EBOOKThis 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.
Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations
Author: Giovanni Bellettini
Publisher: Springer
Published: 2014-05-13
Total Pages: 336
ISBN-13: 8876424296
DOWNLOAD EBOOKThe 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
Author: Theodora Bourni
Publisher: Walter de Gruyter GmbH & Co KG
Published: 2020-12-07
Total Pages: 149
ISBN-13: 3110618362
DOWNLOAD EBOOKWith contributions by leading experts in geometric analysis, this volume is documenting the material presented in the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, on May 29 - June 1, 2018. The central topic of the 2018 lectures was mean curvature flow, and the material in this volume covers all recent developments in this vibrant area that combines partial differential equations with differential geometry.
Mean Curvature Flow and Isoperimetric Inequalities
Author: Manuel Ritoré
Publisher: Springer Science & Business Media
Published: 2010-01-01
Total Pages: 113
ISBN-13: 3034602138
DOWNLOAD EBOOKGeometric 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.
A Course in Minimal Surfaces
Author: Tobias Holck Colding
Publisher: American Mathematical Society
Published: 2024-01-18
Total Pages: 330
ISBN-13: 1470476401
DOWNLOAD EBOOKMinimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.
