For any n-dimensional compact Riemannian Manifold $M$ with smooth metric $g$, by employing the heat kernel embedding introduced by B\'erard-Besson-Gallot'94, we intrinsically construct a canonical family of conformal embeddings $C_{ t,k}$: $M\rightarrow\mathbb{ R}^{ q(t)}$, 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.
Categories: math.DG