Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms

Submanifolds with parallel mean curvature vector play important roles in differential geometry, theory of harmonic maps as well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean curvature vector were classified by B. Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean curvature vector in pseudo-Riemannian spheres and pseudo-hyperbolic spaces with arbitrary codimension and arbitrary index. Consequently, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in all pseudo-Riemannian space forms. As an immediate by-product, we obtain the complete classifications of spatial surfaces with parallel mean curvature vector in arbitrary Lorentzian space forms.      

A rigidity theorem for a space-like graph¶of higher codimension

We prove a rigidity theorem for a space-like graph with parallel mean curvature of arbitrary dimension and codimension in pseudo-Euclidean space via properties of its harmonic Gauss map. We also give an estimate of the squared norm of the second fundamental form in terms of the mean curvature and the image diameter under the Gauss map for space-like submanifolds with parallel mean curvature in pseudo-Euclidean space. The estimate also implies the former theorem. Received: 10 December 1999     

Codimension of immersions with parallel pluri-mean curvature

Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.     

Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis — rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.     

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface.     

Classification of marginally trapped Lorentzian flat surfaces in and its application to biharmonic surfaces

A surface in a semi-Riemannian manifold is called marginally trapped if its mean curvature vector field is light-like at each point. In this article, we classify marginally trapped Lorentzian flat surfaces in the pseudo-Euclidean space . As an application, we obtain the complete classification of biharmonic Lorentzian surfaces in with light-like mean curvature vector.     

Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4-d Spaces and Deformations via Ds Hierarchy

Extensions of the generalized Weierstrass representation to generic surfaces in 4-D Euclidean and pseudo-Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey–Stewartson hierarchy. Geometrically, these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the total squared mean curvature) is the simplest of them. Various particular classes of surfaces and their integrable deformations are considered.     

Some results on space-like self-shrinkers

We study space-like self-shrinkers of dimension n in pseudo-Euclidean space R_m^m+n with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.     

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

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