The embedding dimension of Laplacian eigenfunction maps

TL;DR

This paper establishes bounds on the minimal embedding dimension of Riemannian manifolds via Laplacian eigenfunctions, linking geometric properties to embedding complexity and implications for shape registration.

Contribution

It provides upper bounds on the embedding dimension based on geometric bounds and interprets these results for surfaces in three-dimensional space.

Findings

Maximal embedding dimension depends only on dimension, injectivity radius, Ricci curvature, and volume.

For surfaces in R^3, embedding dimension depends on Gaussian curvature, mean curvature, and surface area.

Results have implications for shape registration techniques.

Abstract

Any closed, connected Riemannian manifold $M$ can be smoothly embedded by its Laplacian eigenfunction maps into $\mathbb{R}^m$ for some $m$. We call the smallest such $m$ the maximal embedding dimension of $M$. We show that the maximal embedding dimension of $M$ is bounded from above by a constant depending only on the dimension of $M$, a lower bound for injectivity radius, a lower bound for Ricci curvature, and a volume bound. We interpret this result for the case of surfaces isometrically immersed in $\mathbb{R}^3$, showing that the maximal embedding dimension only depends on bounds for the Gaussian curvature, mean curvature, and surface area. Furthermore, we consider the relevance of these results for shape registration.