Some Results On Geometry Of Submanifolds In Space Forms

This paper contains four chapters. In Chapter one, we investigate the influence of the boundary on the shape of the space-like hypersurface with constant mean curvature or constant scalar curvature in the Lorentz-Minkowski space Ln+1. When the boundary is a sphere, we extend an uniqueness result due to Alias et al. (see [1]) and obtain the following:Theorem 1.1 The only immersed compact spacelike hypersurfaces with constant mean curvature in Ln+l spanning a sphere Sn-1(r) are Bn(r) or hyperbolic caps.The corresponding problem for hypersurface with constant mean curvature in Euclidean space R"+l remains open. In case of n=3 some partial results about Euclidean three-space have recently been obtained by different authors. Adding a condition, we prove:Theorem 1.2 Let M be a hypersurface with constant mean curvature in Rn+1 spanning a sphere Sn-1 (r). If its Gauss image liesin the semisphere, then M is Bn (r) or a hypersphere cap.When the scalar curvature is constant, we establish the following result:Theorem 1.3 The only immersed compact spacelike hypersurfaces with nonzero constant scalar curvature in Zn+1 spanning a sphere Sn-1 (r) are hyperbolic caps.Theorem 1.4 Let M be a hypersurface with non-zero constant scalar curvature in Rn+l spanning a sphere Sn-1(r). If its Gauss image lies in the semisphere, then M is a hypersphere cap.In 1951, H. Hopf proved, by associating a holomorphic 2-differential to each constant mean curvature surfaces in R3, that only compact genus zero surfaces immersed in R3 with constant mean curvature are the round sphere, In Chapter two, we consider the same question about the compact 2-type surfaces of genus zero in S7. A surface in 5" is called to be of 2-type, if its position vector can bedecomposed as follows:By constructing a series of holomorphic differential forms andusing the vanishing theorem, we prove the following unique theorem:Theorem 2.1 The diagonal sum where x1(S),x2(S) are the unit sphere in R3and the minimalVeronese surface in S4 respectively, is the only topological two-sphere of 2-type in S7.Bonnet surfaces in space form R3( ) and space-like or time-like Bonnet surfaces in indefinite space form R31( ) have been classified by Chen and Li in [25,26]. In Chapter 3, we study the similar problem about the surfaces with parallel mean curvature direction in R4(s) or L4. In the case of closed surface, we obtain the following result:Theorem 3.1 Let M be a closed surface with parallel mean curvature direction in R4( ), which admitting an isometric deformation preserving mean curvature. Then its mean curvature must be constant.Locally, we obtain a following classification result:Theorem 3.2 Suppose that M is a fully immersed surface in R4( ) with parallel mean curvature direction and without umbilical points, which admitting an isometric deformation preserving mean curvature. Then there exists a system of isothermal coordinates (s, t),such that I = e2f(s)(ds2 + dt2) where e2f(s) = F2/bH and the angle of rotation from principal curvature frame to the isothermal frame hasan explicit expression. Moreover, the mean curvature H and Gauss curvature K of M satisfyConversely, M satisfying the above condition admits an isometric deformation preserving mean curvature.In the second part of this chapter, we discuss two kinds of time-like surfaces in L4 which admitting isometric deformation preserving mean curvature, according to VH is of space-like or time-like, and obtain theorem 3.3 and 3.4 similar to theorem 3.2.In Chapter 4, we consider the construction of space-like and time-like surfaces in L3 with K-2mH + m2 =+1 where K and H denote respectively the Gauss curvature and the mean curvature of surfaces, and obtain the following line congruence:(1) Given a space-like immersed surface which satisfies (k1-m)(k2-m) = -1, then there exist other space-like surfaces of same type, and their form can be written as:(2) Given a time-like immersed surface r: M L3, which satisfies then there exist other space-like surfa...