Let G be a connected Lie group. We show that all characteristic classes of G are bounded—when viewed in the cohomology of the classifying space of the group G with the discrete topology—if and only if the derived group of the radical of G is simply connected in its Lie group topology. We also give equivalent conditions in terms of stable commutator length and distortion.

We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn’s linearization theorem.

A well-known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface Mn 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.

A well-known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a zero-order pseudodifferential projection on a closed manifold vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in the odd dimensional case an analogous statement holds for the algebra of projective pseudodifferential operators, i.e., the noncommutative residue of a projective pseudodifferential projection vanishes. Our strategy of proof also simplifies the argument in the classical setting. We show that the noncommutative residue factors to a map from the twisted K-theory of the co-sphere bundle and then use arguments from twisted K-theory to reduce this question to a known model case. It can then be verified by local computations that this map vanishes in odd dimensions.

The classification work [5], [8] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair {4, 5}, {6, 9}, or {7, 8} in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities {4, 5} in S19 is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities {6, 9} in S31 is either the inhomogeneous one constructed by Ferus, Karcher, and M¨unzner, or the one that is homogeneous. This classification reveals the striking resemblance between these two rather different types of isoparametric hypersurfaces in the homogeneous category, even though the one with multiplicities {6, 9} is of the type constructed by Ferus, Karcher, and M¨unzner and the one with multiplicities {4, 5} stands alone. The quaternion and the octonion algebras play a fundamental role in their geometric structures. A unifying theme in [5], [8], and the present sequel to them is Serre’s criterion of normal varieties. Its technical side pertinent to our situation that we developed in [5], [8] and extend in this sequel is instrumental. The classification leaves only the case of multiplicity pair {7, 8} open.