  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.
