共查询到20条相似文献,搜索用时 47 毫秒
1.
In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space. 相似文献
2.
Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form , where and F is a positive, strictly monotone and 1‐homogeneous curvature function. In particular this class includes the mean curvature . We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews–McCoy–Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface. 相似文献
3.
This paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black
hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in (J
Diff Geom 59:353–437), represents a level set formulation for the inverse mean curvature flow describing the evolution of
a hypersurface whose normal velocity equals the reciprocal of its mean curvature. We first propose a finite element method
for a regularized flow which involves a small parameter ɛ; a rigorous analysis is presented to study well-posedness and convergence of the scheme under certain mesh-constraints, and
optimal rates of convergence are verified. We then prove uniform convergence of the finite element solution to the unique
weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter ɛ both tend to zero. Computational results are provided to show the efficiency of the proposed finite element method and to
numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow. Numerical studies
are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter ɛ for optimal convergence of the proposed scheme. Finally, a numerical convergence study for another approach recently proposed
by R. Moser (The inverse mean curvature flow and p-harmonic functions. preprint U Bath, 2005) for approximating the inverse mean curvature flow via p-harmonic functions is also included. 相似文献
4.
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. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established. 相似文献
5.
E.V. Petrov 《Differential Geometry and its Applications》2011,29(4):516-532
We obtain criteria for the harmonicity of the Gauss map of submanifolds in the Heisenberg group and consider the examples demonstrating the connection between the harmonicity of this map and the properties of the mean curvature field. Also, we introduce a natural class of cylindrical submanifolds and prove that a constant mean curvature hypersurface with harmonic Gauss map is cylindrical. 相似文献
6.
In this paper we prove that a compact oriented hypersurface of a Euclidean
sphere with nonnegative Ricci curvature and infinite fundamental group is isometric
to an H(r)-torus
with constant mean curvature. Furthermore, we generalize, whithout any
hypothesis about the mean curvature, a characterization of Clifford torus due to
Hasanis and Vlachos.
Received: 19 March 2002 相似文献
7.
Horatio Quadjovie 《Bulletin des Sciences Mathématiques》2004,128(6):447-466
A rotationally symmetric, compact, oriented, connected, uniformly convex hypersurface M0 of , with boundary ∂M0 in a rotationally symmetric cone S, is evolving under volume-preserving mean curvature flow. Then for n?2, we obtain gradient and curvature estimates, leading to long-time existence of the flow, and convergence to a part of a round sphere. 相似文献
8.
Mehmet Erdoğan 《Geometriae Dedicata》1996,61(3):221-225
We give an estimate for the Ricci curvature of a complete hypersurface M in a hyperbolic space H and in a sphere S under the same condition. As its application, we give the condition for unboundedness of a complete hypersurface M. 相似文献
9.
Oliver C. Schnürer Knut Smoczyk 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2003,20(6):3302-1073
We consider the flow of a strictly convex hypersurface driven by the Gauß curvature. For the Neumann boundary value problem and for the second boundary value problem we show that such a flow exists for all times and converges eventually to a solution of the prescribed Gauß curvature equation. We also discuss oblique boundary value problems and flows for Hessian equations. 相似文献
10.
Henrique F. de Lima Joseílson R. de Lima 《Differential Geometry and its Applications》2012,30(1):136-143
The aim of this paper is to study the uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm and such that its fiber has constant sectional curvature. By using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds and supposing a natural comparison inequality between the r-th mean curvatures of the hypersurface and that ones of the slices of the region where the hypersurface is contained, we are able to prove that a such hypersurface must be, in fact, a slice. 相似文献
11.
We consider the motion of hypersurfaces in Riemannian manifolds by their curvature vectors. We show that the Harnack quadratic
is an affine second fundamental form of the space-time track of the hypersurface. Given a solution to the Ricci flow, we show
that with respect to an appropriate metric on space-time, the space-slices evolve by mean curvature flow. This enables us
to identify the Harnack quadratic for the mean curvature flow with the trace Harnack quadratic for the Ricci flow. 相似文献
12.
Jong Ryul Kim 《Comptes Rendus Mathematique》2013,351(11-12):471-475
For an extrinsic symmetric space M in Minkowski space-time, we prove that if M is spacelike with zero mean curvature, then it is totally geodesic and if M is timelike with zero mean curvature, then it is totally geodesic or it is a flat hypersurface. 相似文献
13.
Esther Cabezas-Rivas Carlo Sinestrari 《Calculus of Variations and Partial Differential Equations》2010,38(3-4):441-469
We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere. 相似文献
14.
Naoyuki Koike 《Geometriae Dedicata》1995,54(1):1-11
We define the concept of a curvature netted hypersurface and investigate in what case the hypersurface is covered by a twisted product of spheres (or topological product of spheres). All hypersurfaces in a space form such that the number of mutually distinct principal curvatures is constant (i.e. each principal curvature has constant multiplicity) are curvature netted hypersurfaces. Also, we state some inductive constructions of the hypersurfaces, where we use the discussion related to the tube. 相似文献
15.
We investigate constant mean curvature complete vertical graphs in a warped product, which is supposed to satisfy an appropriated convergence condition. In this setting, under suitable restrictions on the values of the mean curvature and the norm of the gradient of the height function, we obtain rigidity theorems concerning to such graphs. Furthermore, applications to the hyperbolic and Euclidean spaces are given. 相似文献
16.
Xu Cheng 《Archiv der Mathematik》2006,86(4):365-374
We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional
manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in
with finite index must be minimal.
Received: 30 May 2005 相似文献
17.
Huai-Dong Cao Ying Shen Shunhui Zhu 《Calculus of Variations and Partial Differential Equations》1998,7(2):141-157
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski
space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss
map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends
the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is
bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional
spaces.
Received July 4, 1997 / Accepted October 9, 1997 相似文献
18.
Valentina-Mira Wheeler 《Mathematische Zeitschrift》2014,276(1-2):281-298
We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity. 相似文献
19.
Rugang Ye 《Calculus of Variations and Partial Differential Equations》2008,31(4):439-455
In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities. 相似文献
20.
Luis J. Alías Takashi Kurose Gil Solanes 《Differential Geometry and its Applications》2006,24(5):492-502
We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface. 相似文献