On Gunther’s Perturbation Results for Nash’s Isometric Embedding Theorems

Abstract

In this exposition, we will focus on John Nash's Isometric Embedding Problem: What is the lowest dimension d(n) of Euclidean space that a compact Riemannian manifold of dimension n can be isometrically embedded into? First, we set up the underlying perturbation problem and introduce the loss of differentiability issue that arises. Then, we discuss an elegant solution to the loss of differentiability by Matthias Günther discovered in 1987: His solution involves a nice property of the Laplace operator--in particular, that Δ-1 has a well-defined bounded inverse on Hölder spaces. Günther was able to use this to avoid the monstrosities of proving the inverse function theorem of Nash--Moser and to improve Nash's original upper bound of n(3n+11)/2 for the required dimension of the ambient Euclidean space to max{n(n+5)/2, n(n+3)/2+5}.