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Entao Zhao - One of the best experts on this subject based on the ideXlab platform.
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Mean Curvature flow of higher codimension in hyperbolic spaces
Communications in Analysis and Geometry, 2013Co-Authors: Hongwei Xu, Fei Ye, Entao ZhaoAbstract:where H(x, t) is the Mean Curvature vector of Ft(M) and Ft(x) = F (x, t). We call F : M × [0, T ) → F(c) the Mean Curvature flow with initial value F . The Mean Curvature flow was proposed by Mullins [17] to describe the formation of grain boundaries in annealing metals. In [3], Brakke introduced the motion of a submanifold by its Mean Curvature in arbitrary codimension and constructed a generalized varifold solution for all time. For the classical solution of the Mean Curvature flow, most works have been done on hypersurfaces. Huisken [11, 12] showed that if the initial hypersurface in a Riemannian manifold is uniformly convex, then the Mean Curvature flow converges to a round point in finite time. Later, Huisken [13] extend this result to hypersurfaces satisfying a pinching condition in a sphere. Many other beautiful results have been obtained, and there are various approaches to study the Mean Curvature flow of hypersurfaces (see [6, 7], etc.). For the Mean Curvature flow of submanifolds in higher codimension, some special cases have been studied, see [19, 20, 21, 22, 23, 24] etc. for example. Recently, Andrews-Baker [1] proved a convergence theorem for the Mean Curvature flow of closed submanifolds satisfying a pinching condition in the Euclidean space. In [2], Baker proved a convergence result for the Mean Curvature flow of submanifolds in a sphere. In this paper, we study the Mean Curvature flow of closed submanifolds in hyperbolic
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extend Mean Curvature flow with finite integral Curvature
Asian Journal of Mathematics, 2011Co-Authors: Entao ZhaoAbstract:In this note, we first prove that the solution of Mean Curvature flow on a finite time interval [0,T) can be extended over time T if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain Mean Curvature flow on a finite time interval [0,T) can be extended over time T if the space-time integration of the Mean Curvature is finite. Moreover, we show that these conditions are optimal in some sense.
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Mean Curvature flow of higher codimension in hyperbolic spaces
arXiv: Differential Geometry, 2011Co-Authors: Hongwei Xu, Fei Ye, Entao ZhaoAbstract:In this paper we investigate the convergence for the Mean Curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the Mean Curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.
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the extension for Mean Curvature flow with finite integral Curvature in riemannian manifolds
arXiv: Differential Geometry, 2009Co-Authors: Entao ZhaoAbstract:We investigate the integral conditions to extend the Mean Curvature flow in a Riemannian manifold. We prove that the Mean Curvature flow solution with finite total Mean Curvature on a finite time interval $[0,T)$ can be extended over time $T$. Moreover, we show that the condition is optimal in some sense.
Marcos Dajczer - One of the best experts on this subject based on the ideXlab platform.
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the Mean Curvature of cylindrically bounded submanifolds
arXiv: Differential Geometry, 2008Co-Authors: Luis J Alias, Pacelli G Bessa, Marcos DajczerAbstract:We give an estimate of the Mean Curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant Mean Curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small Mean Curvature.
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constant Mean Curvature hypersurfaces in warped product spaces
Proceedings of the Edinburgh Mathematical Society, 2007Co-Authors: Luis J Alias, Marcos DajczerAbstract:We study hypersurfaces of constant Mean Curvature immersed into warped product spaces of the form $\mathbb{R}\times_\varrho\mathbb{P}^n$, where $\mathbb{P}^n$ is a complete Riemannian manifold. In particular, our study includes that of constant Mean Curvature hypersurfaces in product ambient spaces, which have recently been extensively studied. It also includes constant Mean Curvature hypersurfaces in the so-called pseudo-hyperbolic spaces. If the hypersurface is compact, we show that the immersion must be a leaf of the trivial totally umbilical foliation $t\in\mathbb{R}\mapsto\{t\}\times\mathbb{P}^n$, generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant Mean Curvature hypersurfaces with boundary in a leaf.
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constant Mean Curvature hypersurfaces in warped product spaces
Proceedings of the Edinburgh Mathematical Society, 2007Co-Authors: Luis J Alias, Marcos DajczerAbstract:We study hypersurfaces of constant Mean Curvature immersed into warped product spaces of the form , generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant Mean Curvature hypersurfaces with boundary in a leaf.
Luis J Alias - One of the best experts on this subject based on the ideXlab platform.
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the Mean Curvature of cylindrically bounded submanifolds
arXiv: Differential Geometry, 2008Co-Authors: Luis J Alias, Pacelli G Bessa, Marcos DajczerAbstract:We give an estimate of the Mean Curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant Mean Curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small Mean Curvature.
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constant Mean Curvature hypersurfaces in warped product spaces
Proceedings of the Edinburgh Mathematical Society, 2007Co-Authors: Luis J Alias, Marcos DajczerAbstract:We study hypersurfaces of constant Mean Curvature immersed into warped product spaces of the form $\mathbb{R}\times_\varrho\mathbb{P}^n$, where $\mathbb{P}^n$ is a complete Riemannian manifold. In particular, our study includes that of constant Mean Curvature hypersurfaces in product ambient spaces, which have recently been extensively studied. It also includes constant Mean Curvature hypersurfaces in the so-called pseudo-hyperbolic spaces. If the hypersurface is compact, we show that the immersion must be a leaf of the trivial totally umbilical foliation $t\in\mathbb{R}\mapsto\{t\}\times\mathbb{P}^n$, generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant Mean Curvature hypersurfaces with boundary in a leaf.
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constant Mean Curvature hypersurfaces in warped product spaces
Proceedings of the Edinburgh Mathematical Society, 2007Co-Authors: Luis J Alias, Marcos DajczerAbstract:We study hypersurfaces of constant Mean Curvature immersed into warped product spaces of the form , generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant Mean Curvature hypersurfaces with boundary in a leaf.
Ben Andrews - One of the best experts on this subject based on the ideXlab platform.
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harmonic Mean Curvature flow and geometric inequalities
Advances in Mathematics, 2020Co-Authors: Ben AndrewsAbstract:Abstract We employ the harmonic Mean Curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove Alexandrov-Fenchel type inequalities relating quermassintegrals to the total Curvature, which is the integral of Gaussian Curvature on the hypersurface. The resulting inequality allows us to use the inverse Mean Curvature flow to prove Alexandrov-Fenchel inequalities between the total Curvature and the area for strictly convex hypersurfaces. Finally, we apply the harmonic Mean Curvature flow to prove a new class of geometric inequalities for h-convex hypersurfaces in hyperbolic space.
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harmonic Mean Curvature flow and geometric inequalities
arXiv: Differential Geometry, 2019Co-Authors: Ben AndrewsAbstract:In this article, we will use the harmonic Mean Curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total Curvature, which is the integral of Gaussian Curvature on the hypersurface. We will also use the harmonic Mean Curvature flow to prove a new class of geometric inequalities for horospherically convex hypersurfaces in hyperbolic space. Using these new Alexandrov-Fenchel type inequalities and the inverse Mean Curvature flow, we obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in hyperbolic space, which was previously proved for horospherically convex hypersurfaces by Wang and Xia [44]. Finally, we use the Mean Curvature flow to prove a new Heintze-Karcher type inequality for hypersurfaces with positive Ricci Curvature in hyperbolic space.
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embedded constant Mean Curvature tori in the three sphere
arXiv: Differential Geometry, 2012Co-Authors: Ben AndrewsAbstract:We prove that any constant Mean Curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the Mean Curvature.
Bangyen Chen - One of the best experts on this subject based on the ideXlab platform.
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submanifolds with parallel Mean Curvature vector in riemannian and indefinite space forms
arXiv: Differential Geometry, 2013Co-Authors: Bangyen ChenAbstract:A submanifold of a pseudo-Riemannian manifold is said to have parallel Mean Curvature vector if the Mean Curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel Mean Curvature vector are important since they are critical points of some natural functionals. In this paper, we survey some classical and recent results on submanifolds with parallel Mean Curvature vector. Special attention is paid to the classification of space-like and Lorentz surfaces with parallel Mean Curvature vector in Riemannian and indefinite space forms.
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complete classification of lorentz surfaces with parallel Mean Curvature vector in arbitrary pseudo euclidean space
Kyushu Journal of Mathematics, 2010Co-Authors: Bangyen ChenAbstract:Surfaces with parallel Mean Curvature vector play important roles in the theory of harmonic maps, differential geometry as well as in physics. Surfaces with parallel Mean Curvature vector in Riemannian space forms were classified in the early 1970s by Chen and Yau. Recently, space-like surfaces with parallel Mean Curvature vector in arbitrary indefinite space forms were completely classified by Chen in two papers in 2009. In this paper, we completely classify Lorentz surfaces with parallel Mean Curvature vector in a pseudo-Euclidean space Ems with arbitrary dimension m and arbitrary index s. Our main result states that there are 23 families of Lorentz surfaces with parallel Mean Curvature vector in a pseudo-Euclidean m-space Ems . Conversely, every Lorentz surface with parallel Mean Curvature vector in Ems is obtained from the 23 families.
