Annales Academię Scientiarum Fennicę

Mathematica

Volumen 35, 2010, 131-174

# UNIVERSAL LOCAL PARAMETRIZATIONS VIA HEAT KERNELS AND
EIGENFUNCTIONS OF THE LAPLACIAN

## Peter W. Jones, Mauro Maggioni and Raanan Schul

Yale University, Department of Mathematics

10 Hillhouse Ave, New Haven, CT 06510, U.S.A.; jones 'at' math.yale.edu

Duke University, Department of Mathematics

Box 90320, Durham, NC 27708, U.S.A.; mauro.maggioni 'at' duke.edu

Stony Brook University, Department of Mathematics

Stony Brook, NY 11794-3651, U.S.A.; schul 'at' math.sunysb.edu

**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 *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.

**2000 Mathematics Subject Classification:**
Primary 58J65, 35P99.

**Key words:**
Heat kernel bounds, eigenfunction bounds, local charts,
distortion estimates, bi-Lipschitz mappings, non-linear dimension reduction.

**Reference to this article:** P.W. Jones, M. Maggioni and R. Schul:
Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian.
Ann. Acad. Sci. Fenn. Math. 35 (2010), 131-174.

Full document as PDF file

doi:10.5186/aasfm.2010.3508

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