We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation $F$ of a compact Riemannian manifold $M$ with coefficients in a leafwise $U(p, q)$-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated.

Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold $M$ are established in terms of the isocapacitary function of $M$. The relevant capacity takes a different form according to whether $M$ has finite or infinite volume. Conditions involving the more standard isoperimetric function of $M$ can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples.

A Schubert class $\sigma_\lambda$ in the Grassmannian $G(k, n)$ is rigid if the only proper subvarieties representing that class are Schubert varieties $\Sigma_\lambda$. We prove that a Schubert class $\sigma_\lambda$ is rigid if and only if it is defined by a partition $\lambda$ satisfying a simple numerical criterion. If $\lambda$ fails this criterion, then there is a corresponding hyperplane class in another Grassmannian $G(k', n')$ such that the deformations of the hyperplane in $G(k', n')$ yield non-trivial deformations of the Schubert variety $\Sigma_\lambda$. We also prove that, if a partition λ contains a sub-partition defining a rigid and singular Schubert class in another Grassmannian $G(k', n')$, then there does not exist a smooth subvariety of $G(k, n)$ representing $\sigma_\lambda$.

We prove that if a complete, properly embedded, finite-topology minimal surface in $\mathbf{S}^2 \times \mathbf{R}$ contains a line, then its ends are asymptotic to helicoids, and that if the surface is an annulus, it must be a helicoid.

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less than 2$\pi$ plays an important role in many problems in low-dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli space of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Weiß when the cone angles are less than $\pi$. We prove here an infinitesimal rigidity result valid for cone angles less than 2$\pi$ , stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case. The method is to treat this as a problem concerning the deformation theory of singular Einstein metrics, and to apply analytic methods about elliptic operators on stratified spaces. This work is an important ingredient in the local deformation theory of cone-manifolds by the second author; see also the concurrent work by Weiß.