Isometric embeddings via heat kernel

TL;DR

This paper constructs a canonical family of isometric embeddings for compact Riemannian manifolds using heat kernel perturbations, analyzing their asymptotic geometry as the parameter approaches zero.

Contribution

It introduces a new method for isometric embeddings by perturbing heat kernel embeddings, providing a canonical family parameterized by t.

Findings

Constructed a family of embeddings I_t for small t

Embedded images' asymptotic geometry analyzed as t approaches zero

Embeddings are intrinsically perturbed from heat kernel embeddings

Abstract

For any n-dimensional compact Riemannian manifold (M,g), we construct a canonical t-family of isometric embeddings I_{t}: M->R^{q(t)}, with t>0 sufficiently small and q(t)>>t^{-n/2}. This is done by intrinsically perturbing the heat kernel embedding introduced in [BBG]. As t->0, asymptotic geometry of the embedded images is discussed.