Isometric embeddings via heat kernel

Abstract

For any $n$-dimensional compact Riemannian manifold $M$ with smooth metric $g$, we construct a canonical $t$-family of isometric embeddings $I_t : M \to \mathbb{R}^{q(t)}$, with $t \gt 0$ sufficiently small and $q(t) \gg t^{-\frac{n}{2}}$. This is done by intrinsically perturbing the heat kernel embedding introduced in [BBG]. As $t \to 0_{+}$, asymptotic geometry of the embedded images is discussed.