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Showing 1 to 20 of 45 for “"Mean Curvature"”.
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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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Relating diameter and mean curvature for varifolds
… a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The thesis presents an improved estimate of the set where the lower density is small in terms of the one dimensional …
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The Geometry Of Capillary And Constant Mean Curvature Surfaces
Constant mean curvature (CMC) surfaces are critical points of the area functional for variations that preserve the volume of the region enclosed by the surface. Capillary surfaces are defined in a similar way, but instead of the area functional, one considers a functional that is the sum of the …
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Constant Mean Curvature 1/2 Surfaces in H2 × R
This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc 1/2 surfaces in the three-manifold H 2 × R. The value 1/2 is the critical mean curvature for H 2 × R, in that there do no exist closed cmc surfaces with mean curvature 1/2 or less. Daniel and Hauswirth …
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A New Family of Symmetric Constant Mean Curvature Surfaces
… work, the author studies genus zero constant mean curvature (CMC) surfaces, Alexandrov-embedded in R3. In the first chapters, the work of Lawson, and also of Pinkall and Polthier is extended and the first-order description of correspondence is obtained between constant mean curvature surfaces …
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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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Geometry and analysis of Ricci curvature and mean curvature flows
… the geometry and analysis of spaces with Ricci curvature bounded below from the following three perspectives and the asymptotically conical singularities of mean curvature flows in the following two perspectives. For the spaces with Ricci curvature bounded below, firstly we study the unique …
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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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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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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
… hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at their limiting behaviour as …
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Lorentz Invariant Spacelike Surfaces of Constant Mean Curvature in Anti-de Sitter 3-Space
… invariant spacelike surfaces with constant mean curvature <em>H</em><em> </em>= <em>c</em><em> </em>in the anti-de Sitter 3-space H<sup>3</sup><sub>1</sub>(<em>−c</em><sup>2</sup>) of constant curvature <em>−c</em><sup>2</sup>. In particular, I construct Lorentz invariant spacelike surfaces …
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Self-shrinkers of mean curvature flow and harmonic map heat flow with rough boundary data
In this thesis, first, joint with Longzhi Lin, 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 …
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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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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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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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Analysis of the effects of sire and age on wool quality traits in Romney ewes
… two tooths. Sampling date significantly affected mean fibre diameter (MFD), mean curvature, coarse edge (% fleece 10µ > x̄) and the proportion of ewes with wool medullation (p=<0.001). Average changes in wool traits between hoggets and two tooths were as follows. MFD increased 5.5 micron (hogget …
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