Let $V$ be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on $V$ are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many $V$ the space of complete nonnegatively curved metrics has infinite higher homotopy groups.

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.

For any $H \in [0, 1)$, we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature $H$ embedded in hyperbolic three-space.

We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least $4$ times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the $4$ vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical $4$ vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects.

We study the size of the isometry group $\mathrm{Isom}(M,g)$ of Riemannian manifolds $(M,g)$ as $g$ varies. For $M$ not admitting a circle action, we show that the order of $\mathrm{Isom}(M,g)$ can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of $M$. This generalizes results known for negative Ricci curvature to all manifolds.

More generally we establish a similar universal bound on the index of the deck group $\pi_1 (M)$ in the isometry group $\mathrm{Isom}(\widetilde{M},\widetilde{g})$ of the universal cover $\widetilde{M}$ in the absence of suitable actions by connected groups. We apply this to characterize locally symmetric spaces by their symmetry in covers. This proves a conjecture of Farb andWeinberger with the additional assumption of bounds on curvature, diameter, and injectivity radius. Further we generalize results of KazhdanñMargulis and Gromov on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, G is a compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^{*}$ and the flag manifold structure of $\mathrm{Ad} (G) \xi$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo-Riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $\xi$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.