We consider contact manifolds equipped with Carnot-Caratheodory metrics, and show that the Rumin complex is respected by Sobolev mappings: Pansu pullback induces a chain mapping between the smooth Rumin complex and the distributional Rumin complex. A
s a consequence, the Rumin flat complex -- the analog of the Whitney flat complex in the setting of contact manifolds -- is bilipschitz invariant. We also show that for Sobolev mappings between general Carnot groups, Pansu pullback induces a chain mapping when restricted to a certain differential ideal of the de Rham complex. Both results are applications of the Pullback Theorem from our previous paper.
We show that in an $m$-step Carnot group, a probability measure with finite $m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a polynomial in the moments of the measure, with respect to exponential coordinates. Using this, we
improve the main technical result of our previous paper concerning Sobolev mappings between Carnot groups; as a consequence, a number of rigidity and structural results from that paper hold under weaker assumptions on the Sobolev exponent. We also give applications to quasiregular mappings, extending earlier work in the $2$-step case to general Carnot groups.
We consider mappings $f:Gsupset Urightarrow G$ where $G$ and $G$ are Carnot groups and U is an open subset. We prove a number of new structural results for Sobolev (in particular quasisymmetric) mappings, establishing (partial) rigidity or (partial)
regularity theorems, depending on the context. In particular, we prove the quasisymmetric rigidity conjecture for Carnot groups which are not rigid in the sense of Ottazzi-Warhurst.
We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.
For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of Heinonen-Koskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.
We prove metric differentiation for differentiability spaces in the sense of Cheeger. As corollaries we give a new proof that the minimal generalized upper gradient coincides with the pointwise Lipschitz constant for Lipschitz functions on PI spaces,
a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith is equal to $1$, and new nonembeddability results.
In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description
of singularities and of high curvature regions in a mean convex flow. In the present paper, we give a new treatment of the theory of mean convex (and k-convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem. Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal. A key ingredient in our new approach is the new non- collapsing result of Andrews. Some parts are also inspired by the work of Perelman. In a forthcoming paper, we will give a new construction of mean curvature flow with surgery based on the theorems established in the present paper.
We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions o
f continuous l_p-cohomology, thereby obtaining information about the l_p-equivalence relation, as well as critical exponents associated with l_p-cohomology. As an application, we provide a flexible construction of hyperbolic groups which do not have the Combinatorial Loewner Property, extending and complementing earlier examples. Another consequence is the existence of hyperbolic groups with Sierpinski carpet boundary which have conformal dimension arbitrarily close to 1. In particular, we answer questions of Mario Bonk and John Mackay.
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincare inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new ch
aracterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.
In this note we observe that the notion of an induced representation has an analog for quasi-actions. We then use induced quasi-actions to refine some earlier rigidity results for product spaces.
