Raanan Schul

Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels

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 𝒞α metric). These coordinates are bi-Lipschitz on large neighborhoods of the domain or manifold, with constants controlling the distortion and the size of the neighborhoods that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on novel estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient, that hold in the non-smooth category, and are stable with respect to perturbations within this category. Finally, these coordinate systems are intrinsic and efficiently computable, and are of value in applications.

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 so by extending and improving results of Peter Jones and Kate Okikiolu for sets in ℝd. Their results formed the basis of quantitative rectifiability in ℝd. We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales aroundmost points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analogue of the Square function. This allows us to conclude that for a given set K, the ℓ2 norm of this quantity (which is a function of K) has size comparable to a shortest (Hausdorff length) connected set containing K. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90, Oki92] are independent of the ambient dimension.

Multiscale analysis of 1-rectifiable measures: necessary conditions

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in

How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set

\mathbb {R}^n

Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian

Rn,

AHLFORS-REGULAR CURVES IN METRIC SPACES

n\ge 2