【作者】 唐亮；

【导师】 方丽菁； 欧业林；

【作者基本信息】 广西民族大学 ， 基础数学， 2010， 硕士

【摘要】 双调和映射是黎曼流形之间的双能量泛函的临界点映射.它是调和映射的推广.自姜国英在文[1]中给出双调和映射的方程以来,双调和映射的研究吸引了世界上越来越多的数学家,成为了一个有趣的研究领域.它的一个重要研究方向是双调和子流形（即子流形的包含映射是双调和映射）.在欧氏空间、球面空间、双曲空间中得到了很多关于双调和子流形的构造方法和分类的结果.然而,下列猜想仍未解决:Chen猜想:欧氏空间中的双调和子流形都是极小的.广义的Chen猜想:非正截面曲率的黎曼流形中的双调和子流形都是极小的.在本文,我们研究共形平坦空间中的双调和超曲面,主要工作包括以下几点:首先,我们利用文[2]的黎曼流形中的双调和超曲面方程推导出了共形平坦空间中的双调和超曲面方程.其次,我们给出了所得的方程在以下几个方面的应用:（1）获得了共形平坦空间中的全脐双调和超曲面方程和常平均曲率双调和超曲面方程;（2）将所得方程应用于球面空间和双曲空间,我们重新获得了R.Caddeo,S.Montaldo,C.Oniciuc用别的方法获得的常曲率空间中的双调和超曲面方程,也获得了:Sm（（?））是Sm+1中的非平凡的双调和超曲面;（3）确定了共形度量h =f^-2（z）（dx²+dy²+dz²）,使得（R³, h）中的线性函数的图所决定的平面是非平凡的双调和超曲面;（4）给出了S^m×R中的双调和超曲面方程,也获得了:S^m-1（?）×R是S^m×R中的非平凡的双调和超曲面和一些其它的结果.更多还原

【Abstract】 Biharmonic maps are maps between Riemannian manifolds which are critical pointsof the bi-energy functional. They are generalizations of harmonic maps. Since thederivation of the biharmonic equations appeared in [1], the study of biharmonic mapshas attracted more and more attention of mathematicians in the world and has becomea fascinating area of research. One of the central topics in this area is the study of bi-harmonic submanifolds （i.e., those submanifolds whose inclusion maps are biharmonic）.Many results about constructions and classification of submanifolds in Euclidean spaces,spheres, hyperbolic space forms have been obtained. However, the following conjecturesremain open:Chen’s conjecture: any biharmonic submanifold in Euclidean space Rm is mini-mal.The generalized Chen’s conjecture: any biharmonic submanifold in a space ofnon-positive sectional curvature is minimal.In this thesis, we study biharmonic hypersurfaces in a conformally ?at space. Ourmain work includes the following:First, by using the equation of biharmonic hypersurfaces in Riemannian manifoldsin [2], we deduced the equation of biharmonic hypersurfaces in a conformally ?at space.Next, we give applications of the equation in the following areas:（1）we obtained the equation of constant mean curvature biharmonic hypersurfacesand totally umbilical biharmonic hypersurfaces in a conformally ?at space ;（2）Using the equation in spheres and hyperbolic space forms, we recovered theequation of biharmonic hypersurfaces in constant curvature spaces which R. Caddeo,S. Montaldo, C.Oniciuc obtained by other means. We also recovered the biharmonichyperplane Sm（?） is a proper biharmonic hypersurface in Sm+1 ;（3）Identified a family of conformally ?at metric h = f^-2（z）（dx² +dy² +dz²）, whichturn the graphs of a linear function in （R³, h） into a proper biharmonic surface;（4）We obtained the equation of biharmonic hypersurfaces in S^m×R. We alsoobtained S^m-1（ ?）×R is a proper biharmonic hypersurface in S^m×R and some otherresults.更多还原