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Existence of mean curvature flow singularities with bounded mean curvature [PDF]
Duke Mathematical Journal, 2020 In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $\mathbb{R}^N$ in every dimension $N \ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up.
M. Stolarski
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Mean curvature flow in a Ricci flow background [PDF]
Communications in Mathematical Physics, 2012 Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow.
B. Kleiner +14 more
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A remark on soliton equation of mean curvature flow [PDF]
Anais da Academia Brasileira de Ciências, 2004 In this note, we consider self-similar immersions of the mean curvature flow and show that a graph solution of the soliton equation, provided it has bounded derivative, converges smoothly to a function which has some special properties (see Theorem 1.1 ...
Li Ma, Yang Yang
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Self-Expanders of the Mean Curvature Flow [PDF]
Vietnam Journal of Mathematics, 2020 We study self-expanding solutions M m ⊂ ℝ n $M^{m}\subset \mathbb {R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and ...
Knut Smoczyk
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Mean curvature flow of Reifenberg sets [PDF]
Geometry & Topology, 2017 In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $\mathbb{R}^{n+1}$ starting from any $n$-dimensional $(\varepsilon,R)$-Reifenberg flat set with $\varepsilon$ sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and smooth.
Or Hershkovits
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Fattening in mean curvature flow
For each $g\ge 3$, we prove existence of a compact, connected, smoothly embedded, genus-$g$ surface $M_g$ with the following property: under mean curvature flow, there is exactly one singular point at the first singular time, and the tangent flow at the ...
T. Ilmanen, Brian White
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Conformal solitons for the mean curvature flow in hyperbolic space [PDF]
Annals of Global Analysis and Geometry, 2023 In this paper, we study conformal solitons for the mean curvature flow in hyperbolic space Hn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs ...
L. Mari +3 more
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Inverse mean curvature flow and Ricci-pinched three-manifolds [PDF]
Journal für die Reine und Angewandte Mathematik, 2023 Let ( M , g ) (M,g) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying Ric ≥ ε tr ( Ric ) g \mathrm{Ric}\geq\varepsilon\operatorname{tr}(\mathrm{Ric})g for some ε > 0 \varepsilon>0 .
G. Huisken, T. Koerber
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Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces [PDF]
Numerische Mathematik, 2022 An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a ...
C. M. Elliott +2 more
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Weak-strong uniqueness for volume-preserving mean curvature flow [PDF]
Revista matemática iberoamericana, 2022 In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the ...
Tim Laux
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