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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.
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.
* 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 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.
"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 this book, the author gives a comprehensive account of fundamental results on singularities and the asymptotic behavior o.
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.
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.
* 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.
In this work, we study how solutions of certain non-compact geometric flows of fast-diffusion type interact with their asymptotic geometries at infinity. In the first part, we show the long time existence theorem to the inverse mean curvature flow for complete convex non-compact initial hypersurfaces. The existence and behavior of a solution is tied with the evolution of its tangent cone at infinity. In particular, the maximal time of existence can be written in terms of the area ratio between the initial tangent cone at infinity and the flat hyperplane. In the second part, we study the formation of type II singularity for non-compact Yamabe flow. Assuming the initial metric is conformally flat and asymptotic to a cylinder, we show the higher order asymptotics of the metric determines the curvature blow-up rates at the tip in its first singular time. We also show the singularities of such solutions are modeled on rotationally symmetric steady gradient solitons.