The Related Problems Of The Geometry Of Submanifolds

The geometry of submanifolds is an important branch of the Riemannian geometry. Many geometers have payed a lot attention to it for a long time. We study several rele-vant geometric properties of submanifolds through geometric analysis methods. The thesis includes the rigid results of a special class of minimal submanifolds with flat normal bun-dle in Rn+m, when F - stationary maps are from or into the compact convex hypersurfaces, the nonexistence of constant stable F-stationary maps of a functional related to pullback metrics, some results of F-biharmonic submanifolds,f-biharmonic maps and f-biharmonic submanifolds into a Riemannian manifold with non-posotive sectional curvature.The rigid problem is one of the hot research topics in the geometry of submanifolds and can be reflected through pinching theorem. The minimal submanifolds are a very important class of submanifolds and has an important role in differential geometry, so there are many researchers who study the rigid problem of minimal submanifolds. In the second chapter, we select the second fundamental forms of minimal submanifolds to develop the rigid research. We obtain the rigidity of complete δ-super stable minimal submanifolds without flat normal bundle in Rn+m.Submanifolds often have some relations with some corresponding functional and varia-tion, such as the variational problems of minimal submanifolds and area functional, harmonic submanifolds and energy functional. One manifold and the other one often through a map-ping relationship to establish contact, therefore, using related mapping variations to study the stability, existence, Liouville type results and others becomes one of the popular research topics. In the third chapter, we study the problems related to the instability of F-stationary maps, we obtain the results on the instability of F-stationary maps which are from or into the compact convex hypersurfaces in Euclidean space.Studying the harmonic maps is an important research topic in differential geometry. It has a lot of generalizations, such as biharmonic maps, p-harmonic maps, exponential har- monic maps, F-biharmonic maps,f-biharmonic maps and so on. One of the most important problems in the biharmonic theory is Chen’s conjecture, namely:In 1988, B.Y. Chen raised the conjecture:any biharmonic submanifold in Rn is minimal. Then, there are many gener-alized Chen’s conjectures. Yingbo Han and Shuxiang Feng proposed the problem to consider F-biharmonic maps. In the fourth chapter, we consider the F-biharmonic submanifolds. It’s natural to consider the conjecture:any F-biharmonic submanifold in a Riemannian man-ifold with non-positive sectional curvature is minimal. In the fifth chapter, we consider f-biharmonic maps and f-biharmonic submanifolds, it’s natural to consider the conjecture: any f-biharmonic submanifold in a Riemannian manifold with non-positive sectional curva-ture is minimal. We mainly use the division of integral method and the integral estimation method to get some affirmative partial answers to ours conjectures in this two chapters.