Corpus ID: 238531777

Characterising rectifiable metric spaces using tangent spaces

@inproceedings{Bate2021CharacterisingRM,
  title={Characterising rectifiable metric spaces using tangent spaces},
  author={David Bate},
  year={2021}
}
We characterise rectifiable subsets of a complete metric space X in terms of local approximation, with respect to the Gromov–Hausdorff distance, by an n-dimensional Banach space. In fact, if E ⊂ X with Hn(E) < ∞ and has positive lower density almost everywhere, we prove that it is sufficient that, at almost every point and each sufficiently small scale, E is approximated by a bi-Lipschitz image of Euclidean space. We also introduce a generalisation of Preiss’s tangent measures that is suitable… 

References

SHOWING 1-10 OF 28 REFERENCES
Rectifiable metric spaces: local structure and regularity of the Hausdorff measure
We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show the
Tangents and Rectifiability of Ahlfors Regular Lipschitz Differentiability Spaces
We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger in [Ch99]. We show that if an Ahlfors regular Lipschitz differentiability space has charts of
Analysis of and on uniformly rectifiable sets
The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant
Ultralimits of pointed metric measure spaces
The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent
Rigidity of the 1-Bakry–Émery Inequality and Sets of Finite Perimeter in RCD Spaces
This note is dedicated to the study of the asymptotic behaviour of sets of finite perimeter over RCD(K,N) metric measure spaces. Our main result asserts existence of a Euclidean tangent half-space
Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence
Hausdorff $m$ regular and rectifiable sets in $n$-space
The purpose of this paper is to prove the following theorem: If E is a subset of Euclidean n-space and if the m-dimensional Hausdorff density of E exists and equals one Hm almost everywhere in E,
Carnot rectifiability of sub-Riemannian manifolds with constant tangent
We show that if $M$ is a sub-Riemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive
...
1
2
3
...