共查询到20条相似文献,搜索用时 15 毫秒
1.
2.
Jacques Smulevici 《Annales Henri Poincare》2008,9(8):1425-1453
We address the issue of strong cosmic censorship for T
2-symmetric spacetimes with positive cosmological constant. In the case of collisionless matter, we complete the proof of the
C
2 formulation of the conjecture for this class of spacetimes. In the vacuum case, we prove that the conjecture holds for the
special cases where the area element of the group orbits does not vanish on the past boundary of the maximal Cauchy development.
Submitted: February 2, 2008. Accepted: June 12, 2008. 相似文献
3.
In this paper we study the geometry of null cones in smooth Einstein vacuum spacetimes. We provide the L∞ estimate for the trace of the null second fundamental form, as well as estimates for other geometric quantities. This paper is based on the work of Klainerman and Rodnianski [S. Klainerman, I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (3) (2005) 437–529; S. Klainerman, I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16 (1) (2006) 164–229; S. Klainerman, I. Rodnianski, A geometric Littlewood–Paley theory, Geom. Funct. Anal. 16 (1) (2006) 126–163]. 相似文献
4.
Florian Beyer 《偏微分方程通讯》2017,42(8):1199-1248
We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus T3 and, in this class, we solve the singular initial value problem for the Einstein–Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein–Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only. 相似文献
5.
Yawei CHU 《Frontiers of Mathematics in China》2012,7(1):19-27
Let (M
n
, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper,
by employing an elliptic estimation method, we show that (M
n
, g) is a space form if it has sufficiently small L
n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M
n
, g) with positive scalar curvature. 相似文献
6.
Hans Ringström 《Annales Henri Poincare》2006,7(1):1-20
Recently, progress has been made in the analysis of the expanding direction of Gowdy spacetimes. The purpose of the present
paper is to point out that some of the techniques used in the analysis can be applied to other problems. The essential equations
in the case of the Gowdy spacetimes can be considered as a special case of a wider class of variational problems. Here we
are interested in the asymptotic behaviour of solutions to this class of equations. Two particular members arise when considering
the T3-Gowdy symmetric Einstein-Maxwell equations and when considering T3-Gowdy symmetric IIB superstring cosmology. The main result concerns the rate of decay of a naturally defined energy. A subclass
of the variational problems can be interpreted as wave map equations, and in that case one gets the following picture. The
non-linear wave equations one ends up with have as a domain the positive real line in Cartesian product with the circle. For
each point in time, the wave map can thus be seen as a loop in some Riemannian manifold. As a consequence of the decay of
the energy mentioned above, the length of the loop converges to zero at a specific rate.
Communicated by Sergiu Klainerman
submitted 14/02/05, accepted 21/04/05 相似文献
7.
Bang-Yen Chen 《Monatshefte für Mathematik》1980,90(3):185-194
A surfaceM in a Riemannian manifold is said to have parallel normalized mean curvature vector if the mean curvature vector is nonzero and the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. In this paper, it is proved that every analytic surface in a euclideanm-spaceE
m
with parallel normalized mean curvature vector must either lies in aE
4 or lies in a hypersphere ofE
m
as a minimal surface. Moreover, it is proved that if a Riemann sphere inE
m
has parallel normalized mean curvature vector, then it lies either in aE
3 or in a hypersphere ofE
m
as a minimal surfaces. Applications to the classification of surfaces with constant Gauss curvature and with parallel normalized mean curvature vector are also given. 相似文献
8.
本文研究了单位球中的数量曲率满足r=aH+b的完备超曲面的问题.利用极值原理的方法,获得了超曲面的一个刚性结果,推广了这一类具有常中曲率或者常数量曲率超曲面的结果. 相似文献
9.
In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L
n
+1 as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike
hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L
n
+1 which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates
for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in
L
n
+1 which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and
of a generalization of it.
Received: 5 July 1999 相似文献
10.
Strong Cosmic Censorship for Surface‐Symmetric Cosmological Spacetimes with Collisionless Matter
下载免费PDF全文
![点击此处可从《纯数学与应用数学通讯》网站下载免费的PDF全文](/ch/ext_images/free.gif)
This paper addresses strong cosmic censorship for spacetimes with self‐gravitating collisionless matter, evolving from surface‐symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature k of the symmetric surfaces and the cosmological constant Λ. With a suitable formulation, the question of strong cosmic censorship is settled in the affirmative if Λ=0 or k≤0, Λ > 0. In the case Λ > 0, k=1, we give a detailed geometric characterization of possible “boundary” components of spacetime; the remaining obstruction to showing strong cosmic censorship in this case has to do with the possible formation of extremal Schwarzschild–de Sitter‐type black holes. In the special case that the initial symmetric surfaces are all expanding, strong cosmic censorship is shown in the past for all k,Λ. Finally, our results also lead to a geometric characterization of the future boundary of black hole interiors for the collapse of asymptotically flat data: in particular, in the case of small perturbations of Schwarzschild data, it is shown that these solutions do not exhibit Cauchy horizons emanating from i + with strictly positive limiting area radius.© 2016 Wiley Periodicals, Inc. 相似文献
11.
Yi Bing SHEN Xiao Hua ZHU 《数学学报(英文版)》2005,21(3):631-642
By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n+1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1+1 are considered. 相似文献
12.
We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations □g?+σ? = 𝒢(?,??) on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting nonstatic boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudo-convex hypersurfaces near the conformal boundary. 相似文献
13.
Bang‐Yen Chen 《Mathematische Nachrichten》2005,278(11):1242-1281
A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of AH . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP 2 as well as in complex hyperbolic plane CH 2. We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP 2 and 41 families in CH 2. All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP 2 or in CH 2 is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the
full Riemannian curvature tensor. In this article, supposing (M
n
, g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that
L\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C
0 bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds.
Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M
n
× [0, T), the curvature tensor stays uniformly bounded on M
n
× [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented. 相似文献
15.
In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating
interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper
bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation
problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC
1, cubicC
2, and quarticC
3 splines on refined grids. 相似文献
16.
In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional
subsets Σ
k
of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention
is given to boundaries of convex bodies inR
3. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider
integral functionals depending on the curvature and the area ofK and on the curvature andH
k
of Σ
k
. 相似文献
17.
In this work we consider a complete submanifold M with parallel mean curvature vector h immersed in a space form of constant sectional curvature c £ 0c\leq 0. If M has finite total curvature and |H|2 > -c|H|^2>-c, we prove that M must be compact. 相似文献
18.
In this paper, we consider a conformal minimal immersion f from S
2 into a hyperquadric Q
2, and prove that its Gaussian curvature K and normal curvature K
⊥ satisfy K + K
⊥ = 4. We also show that the ellipse of curvature is a circle. 相似文献
19.
Let M be a helicoidal surface in E
3, free of points of vanishing Gaussian curvature. Let H be the mean curvature and K
II
the curvature of the second fundamental form. In this note it is shown that the helicoidal surfaces satisfying K
II
=H are locally characterized by constancy of the ratio of the principal curvatures. Moreover it is proved that these helicoidal
surfaces are determined by a first order differential equation.
Research supported by E.E.C. contract CHRX-CT92-0050. 相似文献
20.
In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?n if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?n if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献