Gunther’s Proof of Nash’s Isometric Embedding Theorem

Around 1987 a German mathematician named Matthias Gunther found a new way of obtaining the existence of isometric embeddings of a Riemannian manifold. His proof appeared in [1, 2]. His approach avoids the so-called Nash-Moser iteration scheme and, therefore, the need to prove smooth tame or Moser-type estimates for the inverse of the linearized operator. This simplifies the proof of Nash’s isometric embedding theorem [3] considerably. This is an informal expository note describing his proof. It was originally written, because when I first learned Gunther’s proof, it had not appeared either in preprint or published form, and I felt that everyone should know about it. Moreover, since he is at Leipzig, which at the time was part of East Germany, very few mathematicians in the U.S. knew about him or his proof. Since many still seem to be unaware of Gunther’s proof, even after he gave a talk at the International Congress of Mathematicians at Kyoto in 1990 and published his proof in the proceedings [2], I have updated this note and continue to distribute it. I do, however, encourage you to seek out Gunther’s own presentations of his proof.