Conformal Embeddings via Heat Kernel

TL;DR

This paper constructs a canonical family of conformal embeddings for compact Riemannian manifolds using heat kernel methods, highlighting differences from isometric embeddings and providing a new intrinsic approach.

Contribution

It introduces a novel intrinsic construction of conformal embeddings via heat kernel embedding, expanding the understanding of geometric embeddings beyond isometric cases.

Findings

Constructs conformal embeddings for all small t

Shows differences from isometric embeddings

Provides a canonical family of embeddings

Abstract

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.