The embedding dimension of Laplacian eigenfunction maps

Jonathan Bates

doi:10.1016/j.acha.2014.03.002

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arXiv · 1605.01643

The embedding dimension of Laplacian eigenfunction maps

Jonathan Bates

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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.

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Jonathan Bates. 2016-05-04. The embedding dimension of Laplacian eigenfunction maps. https://doi.org/10.1016/j.acha.2014.03.002

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