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Hyperbolic mean curvature flow
Journal of Differential Equations, 2009 In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations are strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with dimension larger than 4. We derive
He, Chun-Lei, Kong, De-Xing, Liu, Kefeng
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Spacelike Mean Curvature Flow [PDF]
The Journal of Geometric Analysis, 2019 AbstractWe prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space$$\mathbb {R}^{n,m}$$Rn,m, which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the$${{\,\mathrm{G}\
Ben Lambert, Jason D. Lotay
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Mean-Curvature Flow of Voronoi Diagrams [PDF]
Journal of Nonlinear Science, 2014 We study the evolution of grain boundary networks by the mean-curvature flow under the restriction that the networks are Voronoi diagrams for a set of points. For such evolution we prove a rigorous universal upper bound on the coarsening rate. The rate agrees with the rate predicted for the evolution by mean-curvature flow of the general grain boundary
Matt Elsey, Dejan Slepcev
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Communications in Contemporary Mathematics, 2019 The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space [Formula ...
Song, Chong, Sun, Jun
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Gaussian mean curvature flow [PDF]
Journal of Evolution Equations, 2010 10 ...
Borisenko, Alexander A., Miquel, Vicente
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Bulletin of the American Mathematical Society, 2015 Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then
Colding, Tobias +2 more
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Mean Curvature Flow of Mean Convex Hypersurfaces [PDF]
Communications on Pure and Applied Mathematics, 2016 In the last 15 years, White and Huisken‐Sinestrari developed a far‐reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high‐curvature regions in a mean convex flow.In the present paper, we give a ...
Haslhofer, Robert, Kleiner, Bruce
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Existence of mean curvature flow singularities with bounded mean curvature
Duke Mathematical Journal, 2023 44 pages, comments ...
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Uniformly Compressing Mean Curvature Flow [PDF]
The Journal of Geometric Analysis, 2018 Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks.
Wenhui Shi, Dmitry Vorotnikov
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Mean curvature flow of spacelike graphs [PDF]
Mathematische Zeitschrift, 2010 We prove the mean curvature flow of a spacelike graph in $(Σ_1\times Σ_2, g_1-g_2)$ of a map $f:Σ_1\to Σ_2$ from a closed Riemannian manifold $(Σ_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Σ_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\leq K_1$, remains a spacelike graph ...
Li, Guanghan, Salavessa, Isabel M. C.
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