Stability analysis of isometric embeddings

TL;DR

This paper proves that for a short time, one can smoothly deform a family of Riemannian metrics on a closed manifold into a corresponding family of isometric embeddings in Euclidean space, using stability estimates and local perturbation methods.

Contribution

It introduces a time-dependent local perturbation method to construct families of isometric embeddings, extending G"unther's approach to a parametric setting.

Findings

Constructs a smooth family of isometric embeddings for a family of metrics.

Uses stability estimates to control the perturbation process.

Provides a method for short-time deformation of metrics into embeddings.

Abstract

In this work we prove the fact that, for a short time, it is possible to construct a smooth parametrized family of isometric embeddings of an arbitrary smooth parametrized family of Riemannian metrics on a smooth closed manifold into an Euclidean space. In order to prove this statement we work out stability estimates within Matthias G\"unther's local perturbation method to derive a time-dependent local perturbation method around a free isometric embedding. Iteratively, we use this time-dependent local perturbation method to construct the desired family of isometric embeddings.