双调和子流形的几何和分类

In the begining of 1980s, J. Eells and L. Lemaire proposed to study 2-harmonic maps between Riemannian manifolds. With the further development of 2-harmonic maps in recent years, the study of biharmonic submanifolds has been developed greatly and is becoming a very active field in geometry of submanifolds. There are two important conjectures arising: Chen’s conjecture and Generalized Chen’s conjecture on the study of biharmonic submanifolds. Although there are some partial answers to the two conjectures, they are still open in general so far. In this project, we intend to examine the existence, classification problems, explicit constructions of biharmonic submanifolds and some related issues, mainly including: (1) the existence problems and rigidity classification of biharmonic submanifolds in Riemannian space forms, in particular, solving Chen’s conjecture in the case of hypersurfaces; (2) the explicit constructions and classification problems of biharmonic submanifolds in product spaces, in particular, the complete classification of biharmonic surfaces in some cases of low dimensions; (3) the classification and topology of biconservative submanifolds, etc. In this project, we hope to give further answers on Chen’s conjecture and Generalized Chen’s conjecture.

J. Eells和L. Lemaire在上世纪八十年代初提出研究黎曼流形间的2-调和映照。近年来，伴随着2-调和映照研究的深入，双调和子流形的研究得到了蓬勃的发展，已经成为子流形几何的研究热点之一。围绕双调和子流形的研究，存在两个重要的猜想：Chen猜想和广义Chen猜想，目前虽有部分答案，但一般情形尚不清楚。本项目拟考察双调和子流形的存在性、分类问题、显式构造及相关问题。主要研究内容包括：研究黎曼空间型中双调和子流形的存在性问题和刚性分类，特别是解决超曲面情形下的Chen猜想；研究乘积空间中双调和子流形的显式构造和分类问题，特别是完全分类低维情形下的双调和曲面；考察Biconservative子流形的分类和拓扑等。本项目将对Chen猜想和广义Chen猜想给出进一步解答。

J. Eells和L. Lemaire在上世纪八十年代初提出研究黎曼流形间的2-调和映照。近年来，伴随着2-调和映照研究的深入，双调和子流形的研究得到了蓬勃的发展，已经成为子流形几何的研究热点之一。从子流形的角度，双调和子流形是极小子流形的重要推广。本项目考察了双调和超曲面、lambda双调和超曲面的存在性、分类问题以及极小曲面的分类问题等。主要研究内容包括：空间型具有常数量曲率双调和超曲面的存在性问题和分类问题，特别是解决了具有至多6个互异主曲率超曲面情形下的Chen猜想和广义Chen猜想；研究了乘积空间中双调和超曲面的分类问题，特别是给出了乘积空间中双调和旋转曲面的常微分方程组；给出了lambda双调和超曲面的一个分类刻画等。