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Constant mean curvature surfaces [PDF]
Surveys in Differential Geometry, 2016 In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for $H$-laminations and CMC foliations of Riemannian $n ...
Meeks III, William H. +2 more
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The mean curvature at the first singular time of the mean curvature flow [PDF]
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2010 Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. We prove that the
Brakke +26 more
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Curvature estimates for surfaces with bounded mean curvature [PDF]
Transactions of the American Mathematical Society, 2011 Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply.
Bourni, Theodora, Tinaglia, Giuseppe
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Mean curvature flow with obstacles [PDF]
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2012 We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme.
Almeida, Luís +2 more
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The hyperbolic mean curvature flow [PDF]
Journal de Mathématiques Pures et Appliquées, 2007 We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector.
LeFloch, Philippe G., Smoczyk, Knut
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Mean Curvature in the Light of Scalar Curvature [PDF]
Annales de l'Institut Fourier, 2020 We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.
M. Gromov
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Curvature estimates for constant mean curvature surfaces [PDF]
Duke Mathematical Journal, 2019 We have separated the original paper into two parts. This new posting is the first part which is self-contained and deals with extrinsic curvature estimates for embedded nonzero constant mean curvature disks.
H. Meeks III, William +1 more
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Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space [PDF]
Calculus of Variations and Partial Differential Equations, 2021 AbstractGiven a constant $$k>1$$ k > 1 , let Z be the family of round spheres of radius $${{\,\mathrm{artanh}\,}}(k^{-1})$$ artanh
G. Cora, R. Musina
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Hyperbolic mean curvature flow [PDF]
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
Chun-Lei He, Kefeng Liu, Dexing Kong
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