We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups except a spectral gap assumption on their action.

Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We prove the following dichotomy: Either the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors, or, otherwise, the action is measurably isometric, in which case there are at most two factors in the product group.

As a first application, this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group. Another application is a strong restriction for actions of nonlinear groups

We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod $2$ flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow

The positive mass conjecture states that any complete asymptotically flat manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the equality case of the positive mass conjecture states that in the above situation, if the mass is zero, then the Riemannian manifold must be Euclidean space. The positive mass conjecture was proved by R. Schoen and S.-T. Yau for all manifolds of dimension less than $8$ (see [SY]), and it was proved by E. Witten for all spin manifolds [Wi]. In this article, we consider complete asymptotically flat manifolds of nonnegative scalar curvature which are also harmonically flat in an end. We show that, whenever the positive mass theorem holds, any appropriately normalized sequence of such manifolds whose masses converge to zero must have metrics that uniformly converge to the Euclidean metric outside a compact region. This result is an ingredient in a proof, coauthored with H. Bray, of the Riemannian Penrose inequality in dimensions less than $8$ (see [BL])

The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal hypersurface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole (see [HI]). In 1999, Bray extended this result to the general case of multiple black holes using a different technique (see [Br]). In this article, we extend the technique of [Br] to dimensions less than eight. Part of the argument is contained in a companion article by Lee [L]. The equality case of the theorem requires the added assumption that the manifold be spin

We show that the maximal number of singular moves required to pass between any two regularly homotopic plane or spherical curves with at most $n$ crossings grows quadratically with respect to $n$. Furthermore, for any two regularly homotopic curves with at most $n$ crossings, there exists such a sequence of singular moves, satisfying the quadratic bound, for which all curves along the way have at most $n+2$ crossings

We prove results toward the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of ${\mathrm{PGL}}_n(\mathbb{R})$. After attaching to each periodic orbit an integral invariant (the discriminant), our results have the following flavor: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most $O({\Delta }^{ϵ})$ orbits of discriminant at most $\Delta$. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We give examples of sequences of periodic orbits of this action that fail to become equidistributed even in higher rank. We also give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees