等参函数及相关问题研究

著名的Yau 猜想断言单位球面中的紧致嵌入极小超曲面的Laplace 算子的第一特征值等于其维数. 近年来有许多几何学家致力于对Yau 猜想的研究, 但是到目前为止, 已有的结论只是一些关于第一特征值估计的不等式. 作为本文的一个主要结果, 本文证明了对于单位球面中的等参极小超曲面,Yau 猜想是正确的. 进一步地, 对于等参超曲面的焦流形(实际上是球面的极小子流形), 本文还证明了在一定维数条件下, 它的第一特征值也是其维数.
作为本文的第二个主要结果, 以著名的Schoen-Yau-Gromov-Lawson 的关于数量曲率的手术理论为出发点, 本文在一个Riemann 流形的嵌入超曲面处作手术, 构造了一个新的具有丰富几何性质的流形, 称为double 流形. 特别地, 本文在单位球面的极小等参超曲面处实行了这一手术, 发现得到的double 流形不仅有很复杂的拓扑(但其示性类有精确描述), 还存在数量曲率为正的度量, 更重要的是保持了等参叶状结构.
比Willmore 曲面更广泛的定义是Willmore 子流形, 即Willmore 泛函在球面中的的极值子流形.单位球面中的Willmore 子流形的例子在已有文献中是非常罕见的. 作为本文的另外两个主要结果, 通过深入挖掘单位球面上的OT-FKM- 型等参函数的焦流形的性质, 本文发现其极大值对应的焦流形是单位球面的一系列Willmore 子流形; 之后, 本文用几何办法统一证明了单位球面中具有4 个不同主曲率的等参超曲面的焦流形都是单位球面的Willmore 子流形. 这些新的Willmore 子流形是极小的,但一般不是Einstein 的.

Isoparametric functions and related topics

The famous Yau conjecture asserts that the rst eigenvalue of every closed minimal embedded hypersurface in the unit sphere is just its dimension. Over several decades, research on the eigenvalues of the Laplace operator has always been a core issue in the study of geometry, many geometricians are committed to the study of Yau conjecture in recent years. However, the results that ever known are only some inequalities on the estimate of the rst eigenvalue. As a main result of this paper, we show that the rst eigenvalue of a closed minimal isoparametric hypersurface in the unit sphere is just its dimension. Furthermore, we show that under some dimensional conditions, the focal submanifolds of an isoparametric hypersurface in the unit sphere also have their dimensions as the rst eigenvalues.
As the second main result of this paper, motivated by the famous Schoen-Yau-Gromov-Lawson surgery theory on scalar curvature, we make a surgery at the embedding hypersurface in a Riemannian manifold, constructing a new manifold with good geometry properties, which is called a double manifold. In particular, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold has complicated topological properties (but its characteristic classes can be precisely described) and carries a metric of positive scalar curvature, more importantly, the isoparametric foliation is kept.
A more extensive de nition of Willmore surface is the Willmore submanifold, which is an extremal submanifold of Willmore functional in spheres. Examples of Willmore submanifolds in the unit sphere are scarce in the literature. As the last two main results of this paper, by taking advantage of isoparametric functions of OT-FKM-type, we give a series of new examples of Willmore submanifolds in the unit sphere; hereafter, we give a uni ed geometric proof that both of focal submanifolds of every isoparametric hypersurface in spheres with four distinct principal curvatures are Willmore. These new examples of Willmore submanifolds are all minimal in spheres, but in general not Einstein.