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

@article{Semmes1996FindingCO,
  title={Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincar{\'e} inequalities},
  author={S. Semmes},
  journal={Selecta Mathematica},
  year={1996},
  volume={2},
  pages={155-295}
}
  • 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… 

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References

SHOWING 1-10 OF 49 REFERENCES

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