  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 $\mathcal{C}^\alpha$ metric). These coordinates
are bi-Lipschitz on embedded balls of the domain or manifold, with distortion
constants that depend only on natural geometric properties of the domain or
manifold. The proof of these results relies on estimates, from above and below,
for the heat kernel and its gradient, as well as for the eigenfunctions of the
Laplacian and their gradient. These estimates 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.
