子流形的刚性及相关问题

The rigidity problem is one of the central topics in the theory of submanifolds.The well-known Chern conjecture and Bernstein problem describe the rigidity phenomena of minimal submanifolds from different angles. Up to now it is still far from a complete solution of each problem. In particular, little is known in the higher codimensional cases. In this project, we shall study the second gap problem for the sectional curvature of closed higher codimensional submanifolds in the Euclidean sphere, based on the works of Peng-Terng etc. Moreover,based on the known works, we shall continue the study of convexity geometry of Grassmannian manifolds, from which one can deduce rigidity theorems on the calibrations. To solve Lawson-Osserman problem (which has a closed relationship with Bernstein problem of higher codimension), the concept of submanifolds with constant Jordan angles will be introduced and the related rigidity problems will be studied. The self-shrinkers (which are closed related to self-similar solutions to the mean curvature flow) and spacelike stationary submanifolds in the Lorentz space are important geometric objects emergencing recently. We shall study their rigidity properity by the known technologies and compare them with minimal submanifolds in Euclidean space. The stability and uniqueness of minimal graphs of higher codimension is also a topic in this subject, which has a closed relationship with the rigidity problem.

刚性问题是子流形理论的核心课题之一.著名的Chern猜想和Bernstein问题从不同角度刻画了极小子流形的"刚性". 迄今为止, 上述两个问题离最终解决还有很远一段距离, 尤其是在高余维情形. 在本项目中, 我们将在Peng-Terng等数学家工作的基础上, 研究球面高余维闭子流形截面曲率的第二空隙问题; 同时, 还将在已有工作的基础上继续研究Grassmann流形的凸几何性质, 并由此推导出与标度相关的刚性定理; 为了研究与高余维Bernstein问题相关的Lawson-Osserman问题,我们提出常Jordan角子流形的概念,并研究与此相关的刚性问题;平均曲率流自收缩子和Lorentz空间中的类空极值子流形是最近出现的重要的几何概念,我们将用已有方法研究它们的"刚性", 并与极小子流形加以比较；最后， 我们还将研究高余维极小图的稳定性和唯一性问题，此问题和刚性问题密切相关.

2015年1月至2018年12月，本课题组在已有文献和项目组成员既有工作的基础上，对子流形的刚性问题进行了系统深入的研究，取得了一系列成果：（1）证明了5维球面中的紧致极小超曲面的一个内刚性定理；（2）对Euclid空间极小子流形的Gauss映照值分布问题进行了深入研究，得到了若干新的Bernstein型定理；（3）引入了“常Jordan角子流形”这一概念，并由此得到了Lawson-Osserman锥的一个刚性定理；（4）进一步发展了Lawson-Osserman锥的构造理论，得到了一系列非参数最小锥的例子；（5）得到了Minkowski空间中的类空极值图的Bernstein型定理；（6）从Gauss映照的角度研究了平均曲率流自收缩子的刚性问题。反映上述成果的9篇论文已发表在Tran. Amer. Math. Soc., Adv. Math., Calc. Var. PDE等SCI杂志上。