共查询到20条相似文献,搜索用时 15 毫秒
1.
《Journal de Mathématiques Pures et Appliquées》2009,91(6):591-614
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
2.
Nikos Frantzikinakis Emmanuel Lesigne Máté Wierdl 《Journal d'Analyse Mathématique》2016,129(1):91-103
We extend the interior gradient estimate due to Korevaar-Simon for solutions of the mean curvature equation from the case of euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary data, thus extending a result due to Dajczer, Hinojosa, and Lira. In addition, we prove the existence and uniqueness of radial graphs in hyperbolic space with prescribed mean curvature function and asymptotic boundary data at infinity. 相似文献
3.
In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is 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 nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow. Moreover, we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski space-time. 相似文献
4.
本文证明了一个拼嵌的爱因斯坦流形中的任何超曲面在沿其平均曲率向量演化时,如果初发始曲面满足保持其截曲率为正的某些条件,则在有限时间内超曲而将收缩成一点。 相似文献
5.
Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds 总被引:1,自引:0,他引:1
Matania Ben-Artzi Philippe G. LeFloch 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2007,24(6):989-1008
Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to non-linear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measure-valued mappings. We establish the well-posedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of geometry-compatible (as we call it) conservation laws is singled out as an important case of interest, which leads to robust Lp estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the L1 contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method. 相似文献
6.
Krzysztof Andrzejewski Pawe? G. Walczak 《Differential Geometry and its Applications》2011,29(6):723-729
In this paper, we study hypersurfaces with constant rth mean curvature Sr. We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with Sr+1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation. 相似文献
7.
邓勤涛 《数学物理学报(B辑英文版)》2011,31(1):353-360
In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4(-1)(H5(-1)) with constant mean curvature H satisfying H2 > 6634 (H2 > 114785 respectively) must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4(-1) (resp. H5(-1)) with constant mean curvature H satisfying H2 > 6643 (resp. H2 > 114785 ) must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9]. 相似文献
8.
In this paper, we investigate initial value problems for hyperbolic mean curvature flow with a dissipative term. By means of support functions of a convex curve, a hyperbolic Monge-Amp`ere equation is derived, and this equation could be reduced to the first order quasilinear systems in Riemann invariants. Using the theory of the local solutions of Cauchy problems for quasilinear hyperbolic systems, we discuss lower bounds on life-span of classical solutions to Cauchy problems for dissipative hyperbolic mean curvature flow. 相似文献
9.
Heiko Kröner 《Mathematische Nachrichten》2014,287(2-3):290-296
In 12 Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in 12 that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface.We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition. 相似文献
10.
本文提出并研究带有线性外力场的双曲平均曲率流,通过凸曲线的支撑函数,导出一个双曲型Monge-Ampère 方程并将其转化成Riemann 不变量满足的拟线性双曲方程组。利用拟线性双曲方程组Cauchy 问题的局部解理论,讨论带有线性外力场的双曲平均曲率流Cauchy 问题经典解的生命跨度(即局部解存在的最大时间区间)。 相似文献
11.
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
12.
We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is
fully non linear and naturally arises in the study of real hypersurfaces in ℂn+1, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real
hypersurface ℂn+1 and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate
elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques
such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem
we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness
of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface. 相似文献
13.
We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing
on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the
volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume
non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum
value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the
manifold is isometric to the Euclidean space. 相似文献
14.
Mark A. S. Aarons 《Calculus of Variations and Partial Differential Equations》2006,25(2):205-246
We study the forced mean curvature flow of graphs in Minkowski space and prove longtime existence of solutions. When the forcing
term is a constant, we prove convergence to either a constant mean curvature hypersurface or a translating soliton – depending
on the boundary conditions at infinity.
It is a pleasure to thank my PhD advisors Klaus Ecker and Gerhard Huisken for their assistance and encouragement. I also thank
Maria Athanassenas, Oliver Schnürrer and Marty Ross for their interest and useful comments, and the Max Planck Gesellschaft
for financial support. 相似文献
15.
We give optimal lower bounds for the hypersurface Diracoperator in terms of the Yamabe number, the energy-momentum tensor andthe mean curvature. In the limiting case, we prove that the hypersurfaceis an Einstein manifold with constant mean curvature. 相似文献
16.
Local well‐posedness for volume‐preserving mean curvature and Willmore flows with line tension
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We show the short‐time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume‐preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non‐local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface. 相似文献
17.
Matthias Makowski 《Calculus of Variations and Partial Differential Equations》2013,46(1-2):213-252
Let N be a (n + 1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface ${\mathcal{S}_{0}}$ and F a curvature function, either the mean curvature H, the root of the second symmetric polynomial ${{\sigma}_{2}=\sqrt{H_{2}}}$ or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature ${{\sigma}_{n}= K^{\frac{1}{n}}}$ . We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the C ∞-topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces. 相似文献
18.
Yan-nan LIU & Huai-yu JIAN Department of Mathematical Sciences Tsinghua University Beijing China 《中国科学A辑(英文版)》2007,50(2):231-239
In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force. 相似文献
19.
Yusuke Higuchi 《Journal of Graph Theory》2001,38(4):220-229
Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001 相似文献
20.
Steven Altschuler Sigurd B. Angenent Yoshikazu Giga 《Journal of Geometric Analysis》1995,5(3):293-358
In this paper, we study generalized “viscosity” solutions of the mean curvature evolution which were introduced by Chen, Giga,
and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally
symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except
at isolated points in spacetime and estimates on the number of such points. 相似文献