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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.
* 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.
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
``Mean curvature flow'' is a term that is used to describe the evolution of a hypersurface whose normal velocity is given by the mean curvature. In the simplest case of a convex closed curve on the plane, the properties of the mean curvature flow are described by Gage-Hamilton's theorem. This theorem states that under the mean curvature flow, the curve collapses to a point, and if the flow is diluted so that the enclosed area equals $\pi$, the curve tends to the unit circle. In thisbook, the author gives a comprehensive account of fundamental results on singularities and the asymptotic behavior of mean curvature flows in higher dimensions. Among other topics, he considers in detail Huisken's theorem (a generalization of Gage-Hamilton's theorem to higher dimension), evolutionof non-convex curves and hypersurfaces, and the classification of singularities of the mean curvature flow. Because of the importance of the mean curvature flow and its numerous applications in differential geometry and partial differential equations, as well as in engineering, chemistry, and biology, this book can be useful to graduate students and researchers working in these areas. The book would also make a nice supplementary text for an advanced course in differential geometry.Prerequisites include basic differential geometry, partial differential equations, and related applications.
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.
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.
In this thesis, we investigate two asymptotic behaviours of the mean curvature flow. The first one is the asymptotic behaviour of singularities of the mean curvature flow, and the asymptotic limit is modelled by the tangent flows. The second one is the asymptotic behaviour of the mean curvature flow as time goes to infinity. We will study several problems related to the asymptotic behaviours. The first problem is the partial regularity of the limit. The partial regularity of mean curvature flow without any curvature assumptions was first studied by Ilmanen. We will follow the idea of Ilmanen to study the partial regularity of other asymptotic limit. In particular, we introduce a generalization of Colding-Minicozzi’s entropy in a closed manifold, which plays a significant role. The second problem is the genericity of the tangent flows of mean curvature flow. The generic mean curvature flow was introduced by Colding-Minicozzi. Furthermore, they introduced mean curvature flow entropy and use it to study the generic tangent flows of mean curvature flow. We study the multiplicity of the generic tangent flow. In particular, we prove that the generic compact tangent flow of mean curvature flow of surfaces has multiplicity 1. This result partially addresses the famous multiplicity 1 conjecture of Ilmanen. One key idea is defining a local version of Colding-Minicozzi’s entropy. We also discuss some related results. These results include a joint work with Zhichao Wang and a joint work with Julius Baldauf.