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The embedding dimension of Laplacian eigenfunction maps
Any closed, connected Riemannian manifold can be smoothly embedded by its
Laplacian eigenfunction maps into for some . We call the
smallest such the maximal embedding dimension of . We show that the
maximal embedding dimension of is bounded from above by a constant
depending only on the dimension of , 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 , 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.Comment: 16 pages, 2 figures, 3 theorems, and a torus in a pear tre