Isoparametric functions and related topics

Abstract

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.