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

Shiv Yajnik — Mon, 29 Jun 2026 00:00:00 +0000

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}.

One Small Step For Stability, One Giant Leap For Schwarzschild: Boundedness of Scalar Waves on Schwarzschild Spacetimes

Katherine Mekechuk — Mon, 29 Jun 2026 00:00:00 +0000

This paper gives a detailed proof of the boundedness of the scalar wave equation on a Schwarzschild spacetime using modern energy estimates. The first proof of this statement was given by Kay and Wald in by using Killing vector fields and discrete isometries of the spacetime. However, since then, new techniques to analyze vector fields near the event horizon have been developed by Dafermos and Rodnianski in which has given rise to a new proof strategy. This new proof, using the red shift effect, has been sketched in many lecture notes, but many details are left out. This paper fills in those details and presents the theorem in a self-contained manner.

Columbia Journal of Undergraduate Mathematics

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

One Small Step For Stability, One Giant Leap For Schwarzschild: Boundedness of Scalar Waves on Schwarzschild Spacetimes