Nash twist and Gaussian noise measure on isometric $C^1$ maps

TL;DR

This paper constructs a sequence of random isometric maps from a starting map, demonstrating that scaled differences converge in distribution to a Gaussian noise measure, using Nash twist techniques.

Contribution

It introduces a novel application of Nash twist to generate random isometric maps whose scaled differences converge to a Gaussian measure.

Findings

Difference $(f_n - f_0)$ vanishes in $C^0$ norm as $n$ increases.

Scaled differences $n^{1/2}(f_n - f_0)$ converge weakly to Gaussian noise.

Method combines isometric embedding with probabilistic convergence analysis.

Abstract

Starting with a short map $f_0:I\to \mathbb R^3$ on the unit interval $I$, we construct random isometric map $f_n:I\to \mathbb R^3$ (with respect to some fixed Riemannian metrics) for each positive integer $n$, such that the difference $(f_n - f_0)$ goes to zero in the $C^0$ norm. The construction of $f_n$ uses the Nash twist. We show that the distribution of $ n^{1/2} (f_n - f_0)$ converges (weakly) to a Gaussian noise measure.