Conformal Embeddings via Heat Kernel

Abstract

For any n-dimensional compact Riemannian Manifold $M$ with smooth metric $g$, by employing the heat kernel embedding introduced by Bérard-Besson-Gallot'94, we intrinsically construct a canonical family of conformal embeddings $C_{t,k}$: $M\to {\mathbb{R}}^{q(t)}$, with $t>0$ sufficiently small, $q(t)\gg t^{-\frac{n}{2}}$, and $k$ as a function of $O(t^l)$ in proper sense. Our approach involves finding all these canonical conformal embeddings, which shows the distinctions from the isometric embeddings introduced by Wang-Zhu'15.

25 pages, LaTeX. Revised version with minor corrections. Accepted for publication in J. Geom. Anal