Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels

• DOI: 10.1073/pnas.0710175104
• Publication Date: 2008-02-08T01:59:26Z
• Authors: Peter W. Jones; Mauro Maggioni; Raanan Schul
• Keywords: Heat kernel; Manifold (fluid mechanics); Riemannian manifold; Eigenfunction; Kernel (algebra); Distortion (music)

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g., with (alpha) metric). These are bi-Lipschitz neighborhoods domain manifold, constants controlling distortion size that depend only natural geometric properties manifold. The proof these results relies novel estimates, from above below, for kernel its gradient, as well their hold in non-smooth category, stable respect perturbations within this category. Finally, coordinate systems intrinsic efficiently computable, value applications.