Universal Local Parametrizations via Heat Kernels and Eigenfunctions of the Laplacian

By Peter W. Jones, Mauro Maggioni, Raanan Schul. 2007

54 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f06c778af0a49bc4b3bc356a954139ce645ba7c4

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In Applied Mathematics

1. Sanjoy K. Mitter
2. Hariharan Narayanan
3. Frank Filbir
4. Hrushikesh Narhar Mhaskar
5. Charles Fefferman
6. Jacobus W. Portegies
7. R. Myers
8. E. Gruyer

In Computational Mechanics

1. Frank Filbir
2. Hrushikesh Narhar Mhaskar
3. Sanjoy K. Mitter
4. Charles Fefferman
5. Hariharan Narayanan
6. Jacobus W. Portegies
7. Jürgen Prestin

In Geometry and Topology

1. Hrushikesh Narhar Mhaskar
2. Charles Fefferman
3. Frank Filbir
4. Jacobus W. Portegies
5. R. Myers
6. E. Gruyer
7. Robin Graham
8. Thomas Leistner

Citations in the corpus

• Testing the Manifold Hypothesis (2013, background): “In the case of the “Laplacian eigenmap” [3,30] again, a nearest neighbor graph is formed.”
• Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions (2013, background): “In the extended version of the paper [30], Jones et al.”
• A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel (2010, background): “On a more theoretical side, Jones, Maggioni, and Schul [18] have recently proved that the heat kernel can be used to construct a local coordinate atlas on manifolds, preserving the order of magnitude of the distances between points within each chart.”
• Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law (2014, unknown): “Maps related to Ψλ are studied in [2, 20, 24, 41].”