We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of the enclosed volume, and we completely describe the isoperimetric surfaces for very large enclosed volume. This complements work in H. Bray’s thesis, where isoperimetric surfaces homologous to the horizon are studied.

In this article we give necessary and sufficient conditions for an irreducible Kähler C-space with $b_2 = 1$ to have nonnegative or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the five exceptional Lie groups $E_6,E_7,E_8,F_4,G_2$, the computer package MAPLE is used to assist our calculations. The results are related to two conjectures of Li-Wu-Zheng.

The classification work Isoparametric hypersurfaces with four principal curvatures, and Isoparametric hypersurfaces with four principal curvatures, II, 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 $S^{19}$ is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities $\{6, 9\}$ in $S^{31}$ is either the inhomogeneous one constructed by Ferus, Karcher, and Münzner, 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 Münzner 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 Isoparametric hypersurfaces with four principal curvatures, and Isoparametric hypersurfaces with four principal curvatures, II, and the present sequel to them is Serre’s criterion of normal varieties. Its technical side pertinent to our situation that we developed in Isoparametric hypersurfaces with four principal curvatures, and Isoparametric hypersurfaces with four principal curvatures, II and extend in this sequel is instrumental.

The classification leaves only the case of multiplicity pair $\{7, 8\}$ open.

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.

A 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.