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Showing 1 to 20 of 22 for “"mean-curvature-flow"”.
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Uniqueness problems in mean curvature flow
We investigate uniqueness phenomena in mean curvature flow, focusing on two central problems: the behavior of the flow near singularities and the extension of the flow beyond singular times. These questions have significant applications in geometry, topology, and analysis. For the first problem, …
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The Morse index of mean curvature flow self-shrinkers
… introduce a notion of index of shrinkers of the mean curvature flow. We will then prove a gap theorem for the index of rotationally symmetric immersed shrinkers in R3, namely, that such shrinkers have index at least 3, unless they are one of the stable ones: the sphere, the cylinder, or the …
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Self-shrinkers and translating solitons of mean curvature flow
We study singularity models of mean curvature flow ("MCF") and their generalizations. In the first part, we focus on rigidity and curvature estimates for self-shrinkers. We give a rigidity theorem proving that any self-shrinker which is graphical in a large ball must be a hyperplane. This result …
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Singular behaviour and long time behaviour of mean curvature flow
… 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 …
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Mean curvature flow self-shrinkers with genus and asymptotically conical ends
… of Minimal Surfaces and of singularities in Mean Curvature Flow, for smooth submanifolds Y" in an ambient Riemannian (n+ 1)-manifold Nn+1, including: (1) New asymptotically conical self-shrinkers with a symmetry, in R"+1. (1') Classification of complete embedded self-shrinkers with a …
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A Survey of Convexity Estimates and Singularities for Mean Curvature Flow with Surgery
In this survey we aim to introduce the basics of Mean Curvature Flow and detail the programme of Mean Curvature Flow with surgery developed by Huisken and Sinestrari over four seminal papers, [29],[28],[30] and [8]. We divide the survey into 3 chapters, the rst being an introduction to the Mean …
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A Survey of Convexity Estimates and Singularities for Mean Curvature Flow with Surgery
In this survey we aim to introduce the basics of Mean Curvature Flow and detail the programme of Mean Curvature Flow with surgery developed by Huisken and Sinestrari over four seminal papers, [29],[28],[30] and [8]. We divide the survey into 3 chapters, the rst being an introduction to the Mean …
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Self-similar solutions to the mean curvature flow in Euclidean and Minkowski space
… self-similar solutions to the curve shortening flow in the Euclidean plane R² and discuss basic properties of the curves. The problem of finding the curves is reduced to the study of a twodimensional system of ODEs with two parameters that determine the type of the self-similar motion. In the …
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Self-shrinkers of mean curvature flow and harmonic map heat flow with rough boundary data
… we establish estimates for the harmonic map heat flow from the unit circle into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a …
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Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry
… solutions to the level set equation for mean curvature flow. We describe a set of hypotheses under which we can prove that the level sets of these solutions are C¹,¹ submanifolds of spacetime with well understood behavior near singular times. We then relate the derivatives of the solution …
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L^\infty Fehlerabschätzungen im Kontext geometrischer Evolutionsgleichungen
… Gleichungen nachgewiesen und auf die mean-curvature-flow Gleichung angewendet. Weiter wird eine Fehlerabschaetzung in L^2 fuer Willmore-flow auf drei Raumdimensionen verallgemeinert.
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Foliations For Quasi-Fuchsian 3-Manifolds
… contains a minimal surface whose principal curvatures are in (-1, 1), then it admits a foliation such that each leaf is a surface of constant mean curvature. The key method that we use here is volume preserving mean curvature flow.
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Negative Point Mass Singularities in General Relativity
… from continuous matter. We review the Inverse Mean Curvature Flow and use this flow to prove some new results about the mass of a singularity, the ADM mass of the manifold, and the capacity of the singularity. We describe some particular examples of these singularities that exhibit additional …
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Structure of Singular Sets Local to Cylindrical Singularities for Stationary Harmonic Maps and Mean Curvature Flows
… singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature …
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Maximal Graphs and Spacelike Mean Curvature Flows in Semi-Euclidean Spaces
… regularity theorem, but now for the spacelike mean curvature flow system in semi-Euclidean spaces. This is proved using a version of Huisken’s monotonicity formula. Under the assumption of a suitable gradient bound, this theorem will give a priori estimates that allow such flows to be smoothly …
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Non-collapsing on Hypersurfaces of Prescribed Curvature
… hypersurfaces with two distinct principal curvatures into $\mathbb{S}^{n+1}$. Also, we assume that these hypersurfaces satisfying a PDE of the principal curvatures with assumptions on the number of multiplicities of these curvatures. As a result, we deduce that these principal curvatures …
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Non-collapsing on Hypersurfaces of Prescribed Curvature
… hypersurfaces with two distinct principal curvatures into $\mathbb{S}^{n+1}$. Also, we assume that these hypersurfaces satisfying a PDE of the principal curvatures with assumptions on the number of multiplicities of these curvatures. As a result, we deduce that these principal curvatures …
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Level set methods for higher order evolution laws
… is given. Four different models are discussed: mean curvature flow, surface diffusion, a kinetic model, which combines the effects of mean curvature flow and surface diffusion and includes a further kinetic component, and an adatom model, which incorporates in addition free adatoms. As an …
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The 1-dimensional [lambda]-self shrinkers in R² and the nodal sets of biharmonic Steklov problems
… problems. Chapter 1 gives the background of mean curvature flow and the importance of self-shrinkers as solitons of the flow equation. We also introduce the background of the [lambda]-hypersurface and explain how this is related to the self shrinkers. In chapter 2, we examine the solutions of …
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