Relative Lorentzian volume comparison with integral Ricci and scalar curvature bound

A relative Lorentzian volume comparison estimate between spacelike hypersurfaces is studied with the integral curvature bound in terms of Ricci and Scalar curvature which generalize the Bishop–Gromov volume comparison theorem.     

Spacelike submanifolds of codimension two in de Sitter space

We investigate the differential geometry of spacelike submanifolds of codimension two in de Sitter space and classify the singularities of lightlike hypersurfaces and lightcone Gauss maps in de Sitter 4-space.     

Complete explicit classification of parallel Lorentz surfaces in arbitrary pseudo-Euclidean spaces

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in physics since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel spacelike surfaces in an arbitrary indefinite space form are classified in Chen (2010) [20]. Moreover, parallel Lorentz surfaces in 4D indefinite space forms are completely classified in a series of recent articles Chen (submitted for publication) [16], Chen (submitted for publication) [17], Chen (in press) [18], Chen (2010) [19], Chen and Van der Veken (2009) [15] (see also Graves (1979) [12], Graves (1979) [13] and Magid (1984) [14] for some partial results). In this paper, we achieve the complete classification of parallel Lorentz surfaces in a pseudo-Euclidean space with arbitrary codimension and arbitrary index.     

On surfaces whose twistor lifts are harmonic sections

We study surfaces whose twistor lifts are harmonic sections, and characterize these surfaces in terms of their second fundamental forms. As a corollary, under certain assumptions for the curvature tensor, we prove that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. For surfaces in four-dimensional Euclidean space, a lower bound for the vertical energy of the twistor lifts is given. Moreover, under a certain assumption involving the mean curvature vector field, we characterize a surface in four-dimensional Euclidean space in such a way that the twistor lift is a harmonic section, and its vertical energy density is constant.     

Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.     

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.     

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

In this paper, the complete spacelike hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. Several classified theorems are obtained using the operator _L1$L_1$ introduced by S.Y. Cheng and S.T. Yau (1977) [4].     

Prescribing singularities of maximal surfaces via a singular Björling representation formula

We derive a proper formulation of the singular Björling problem for spacelike maximal surfaces with singularities in the Lorentz–Minkowski 3-space which roughly asks whether there exists a maximal surface that contains a prescribed curve as singularities, and then provide a representation formula which solves the problem in an affirmative way. As consequences, we construct many kinds of singularities of maximal surfaces and deduce some properties of the maximal surfaces related to the singularities due to the geometry of the Gauss map.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Isolation Phenomenon for Complete Surfaces Arising from Yang–Mills Theory

Let M be a complete surface with parallel mean curvature in a complete simply connected space form F ^2+p (c) of constant curvature c. Denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively. Motivated by L ²-isolation phenomenon in Yang–Mills theory, we prove that if , where c + H ² > 0, D(H,c) is an explicit positive constant depending on H and c, then , i.e., M is a totally umbilical sphere . Research supported by the Chinese NSF, Grant No. 10231010; Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.     

Constant Mean Curvature Spacelike Surfaces in Three-Dimensional Generalized Robertson–Walker Spacetimes

Several uniqueness and non-existence results on complete constant mean curvature spacelike surfaces lying between two slices in certain three-dimensional generalized Robertson–Walker spacetimes are given. They are obtained from a local integral estimation of the squared length of the gradient of a distinguished smooth function on a constant mean curvature spacelike surface, under a suitable curvature condition on the ambient spacetime. As a consequence, all the entire bounded solutions to certain family of constant mean curvature spacelike surface differential equations are found.     

Positivity of quasilocal mass

Motivated by the important work of Brown and York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with a positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface.     

On complete spacelike hypersurfaces with two distinct principal curvatures in Lorentz–Minkowski space

We investigate complete spacelike hypersurfaces in Lorentz–Minkowski space with two distinct principal curvatures and constant m$m$th mean curvature. By using Otsuki’s idea, we obtain the global classification result. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Lorentz–Minkowski (n+1)$(n+1)$-spaces (n≥3$n\ge 3$) of nonzero constant m$m$th mean curvature (m≤n−1$m\le n-1$) with two distinct principal curvatures λ$\lambda$ and μ$\mu$ satisfying inf(λ−μ)²>0$inf{(\lambda -\mu )}^2>0$ are the hyperbolic cylinders. We also obtain a global characterization for hyperbolic cylinder Hⁿ⁻¹(c)×R${\mathbb{H}}^{n-1}\left(c\right)\times \mathbb{R}$ in terms of square length of the second fundamental form.     

The theorem of Schur in the Minkowski plane

We prove a Lorentzian analogue of the theorem of Schur for spacelike (or timelike) curves in the Minkowski plane.     

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Riemannian space form

In this work we obtain a gap theorem for spacelike submanifolds with parallel mean curvature vector in a semi-Riemannian space form.     

Maximal hypersurfaces and foliations of constant mean curvature in general relativity

We prove theorems on existence, uniqueness and smoothness of maximal and constant mean curvature compact spacelike hypersurfaces in globally hyperbolic spacetimes. The uniqueness theorem for maximal hypersurfaces of Brill and Flaherty, which assumed matter everywhere, is extended to spacetimes that are vacuum and non-flat or that satisfy a generic-type condition. In this connection we show that under general hypotheses, a spatially closed universe with a maximal hypersurface must be Wheeler universe; i.e. be closed in time as well. The existence of Lipschitz achronal maximal volume hypersurfaces under the hypothesis that candidate hypersurfaces are bounded away from the singularity is proved. This hypothesis is shown to be valid in two cases of interest: when the singularities are of strong curvature type, and when the singularity is a single ideal point. Some properties of these maximal volume hypersurfaces and difficulties with Avez' original arguments are discussed. The difficulties involve the possibility that the maximal volume hypersurface can be null on certain portions; we present an incomplete argument which suggests that these hypersurfaces are always smooth, but prove that an a priori bound on the second fundamental form does imply smoothness. An extension of the perturbation theorem of Choquet-Bruhat, Fischer and Marsden is given and conditions under which local foliations by constant mean curvature hypersurfaces can be extended to global ones is obtained.     

The Gauss map of minimal surfaces in the Anti-de Sitter space

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric embedding of Anti-de Sitter space as a quadric in R^2,2${\mathbb{R}}^{2,2}$. We prove that the Gauss map of a minimal surface immersed in SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ is harmonic. Conversely, we exhibit a representation of minimal surfaces in Anti-de Sitter space in terms of a given harmonic map.