Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

  title={Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincar{\'e} inequalities},
  author={S. Semmes},
  journal={Selecta Mathematica},
  • S. Semmes
  • Published 1 December 1996
  • Mathematics
  • Selecta Mathematica
In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that are well distributed across a region. In the present paper we give geometric criteria on a metric space under which we can find similar families of curves. We shall find these curves by first solving a “dual” problem of building Lipschitz maps from our metric space into a sphere with good… 
Sobolev Spaces and Quasiconformal Mappings on Metric Spaces
Heinonen and I have recently established a theory of quasiconformal mappings on Ahlfors regular Loewner spaces. These spaces are metric spaces that have sufficiently many rectifiable curves in a
Geometric characterizations of p-Poincaré inequalities in the metric setting
We prove that a locally complete metric space endowed with a doubling measure satisfies an ∞-Poincar´e inequality if and only if given a null set, every two points can be joined by a quasiconvex
Lipschitz-type Sobolev Spaces in Metric Measure Spaces
We compare different Lipschitz-type spaces defined on a metric space. In the case of a metric measure space, we will also compare these spaces with the Newtonian-Sobolev space N(X). If the space is
The ∞-Poincaré Inequality in Metric Measure Spaces
A useful feature of the Euclidean n-space, n ≥ 2, is that every pair of points x and y can be joined not only by the line segment [x, y] but also by a large family of curves whose length is
Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds
We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in Rn. The condition is a Sobolev condition for a measurable coframe of flat
Uniformization of two-dimensional metric surfaces
We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of
We study quasiregular mappings from a generalized n-manifold to R. One motivation for our study arises from the difficult question of deciding when a given metric space is locally bi-Lipschitz
Tangent Lines and Lipschitz Differentiability Spaces
Abstract We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric


Good metric spaces without good parameterizations
A classical problem in geometric topology is to recognize when a topological space is a topological manifold. This paper addresses the question of when a metric space admits a quasisymmetric
Quantitative rectifiability and Lipschitz mappings
The classical notion of rectifiability of sets in R n is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with L p estimates
Some remarks about metric spaces, spherical mappings, functions and their derivatives
If $p \in {\bold R}^n$, then we have the radial projection map from ${\bold R}^n \backslash \{p\}$ onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is
Quasiconformal 4-manifolds
For any pseudo-group of homeomorphism of Euclidean space one can define the corresponding category of manifolds. The most familiar examples in Topology are the full pseudo-group of homeomorphisms,
Topological finiteness theorems for manifolds in Gromov-Hausdorff space
We give general conditions under which precompact sets of topological manifolds in Gromov-Hausdorff space contain finitely many homeomorphism types. The main result says that this is true if the
Subelliptic estimates and function spaces on nilpotent Lie groups
In recent years there has been considerable activity in the study of hypoelliptic but non-elliptic partial differential equations. (We recall that a differential operator ~e on a manifold M is said
On the nonexistence of bilipschitz parameterizations and geometric problems about $A_\infty$-weights
How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of Rn are already quite
Shrinking cell-like decompositions of manifolds. Codimension three
Euclidean n-space En, n > 5, has the following simple DISJOINT DISK PROPERTY: singular 2-dimensional disks in En may be adjusted slightly so as to be disjoint. We show that for a large class of
Quasisymmetric embeddings in Euclidean spaces
We consider quasi-symmetric embeddings/: G -, R", G open in Rp, p < n. If p = n, quasi-symmetry implies quasi-conformality. The converse is true if G has a sufficiently smooth boundary. If p < n, the