Dedicated to Professor Banghe Li on his 70th birthday. AbstractA well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface M n in the unit sphere S n+1 (1) is just its dimension n. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.

This paper is a continuation and wide extension of [TY]. In the first part of the present paper, we give a unified geometric proof that both focal submanifolds of every isoparametric hypersurface in spheres with four distinct principal curvatures are Willmore. In the second part, we completely determine which focal submanifolds are Einstein except one case.2000 Mathematics Subject Classification. 53A30, 53C42.

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Abstract. The focal sets of isoparametric hypersurfaces in spheres with g = 4 are all Willmore submanifolds, being minimal but mostly non-Einstein ([TY1], [QTY]). Inspired by A.Gray's view, the present paper shows that, these focal sets are all Amanifolds but rarely Ricci parallel, except possibly for the only unclassified case. As a byproduct, it gives infinitely many simply-connected examples to the problem 16.56 (i) of Besse concerning generalizations of the Einstein condition.

This paper introduces the notion of k-isoparametric hypersurface in an (n + 1)-dimensional Riemannian manifold for k = 0, 1, ..., n. Many fundamental and interesting results ( towards the classification of homogeneous hypersurfaces among other things ) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces.2010 Mathematics Subject Classification. 53C42, 53C24.