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Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels
TLDR
This work uses heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds that hold in the non-smooth category, and are stable with respect to perturbations within this category.
Subsets of rectifiable curves in Hilbert space-the analyst’s TSP
We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do
An analyst’s traveling salesman theorem for sets of dimension larger than one
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $$\beta $$β-numbers. These $$\beta $$β-numbers are geometric quantities
Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps
We prove a global implicit function theorem. In particular we show that any Lipschitz map $${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$$ (with n-dim. image) can be
Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space
We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of
Universal Local Parametrizations via Heat Kernels and Eigenfunctions of the Laplacian
We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with
The traveling salesman problem in the Heisenberg group: upper bounding curvature
We show that if a subset $K$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's
Quantitative decompositions of Lipschitz mappings into metric spaces
We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which
A doubling measure on R^d can charge a rectifiable curve
For d > 2, we construct a doubling measure v on ℝ d and a rectifiable curve Γ such that ν(Γ) > 0.
Multiscale Analysis of 1-rectifiable Measures II: Characterizations
Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean
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