The embedding dimension of Laplacian eigenfunction maps
This work addresses a foundational problem in differential geometry and shape analysis, with potential applications in shape registration, but it appears incremental as it builds on existing embedding theory by providing new bounds.
The authors tackled the problem of determining the minimal dimension needed to embed Riemannian manifolds using Laplacian eigenfunction maps, showing that this maximal embedding dimension is bounded by constants related to geometric properties like dimension, injectivity radius, Ricci curvature, and volume, with specific bounds derived for surfaces in terms of Gaussian curvature, mean curvature, and area.
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.