Embeddings of Riemannian manifolds with heat kernels and eigenfunctions

Abstract

We show that any closed n-dimensional Riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points. Both this time and this number can be bounded in terms of the dimension, a lower bound on the Ricci curvature, the injectivity radius and the volume. It follows that the manifold can be embedded by a finite number of eigenfunctions of the Laplace operator. Again, this number only depends on the geometric bounds and the dimension. In addition, both maps can be made arbitrarily close to an isometry. In the appendix, we derive quantitative estimates of the harmonic radius, so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well.