Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels

Abstract

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 (alpha) 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.