The embedding dimension of Laplacian eigenfunction maps

05/04/2016

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by   Jonathan Bates, et al.

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Any closed, connected Riemannian manifold M can be smoothly embedded by its Laplacian eigenfunction maps into 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 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.