A Measure on the space of Lipschitz isometric maps of a compact 1-manifold into $\mathbb R^2$

TL;DR

This paper constructs and analyzes a probability measure on the space of Lipschitz isometric maps of a compact 1-manifold into the plane, using Nash's convergence technique to study solutions of a differential equation defining these maps.

Contribution

It introduces a novel probability measure on the space of almost everywhere differentiable isometric immersions of 1-manifolds into a52, leveraging Nash's method.

Findings

Established a probability measure on the space of solutions.

Analyzed the properties of this measure in the context of isometric immersions.

Connected Nash's convergence technique to the construction of measures.

Abstract

Let $M$ be a compact 1-manifold. Given a continuous function $g:M\to \mathbb R_+$ we consider the following ordinary differential equation: $\|\dot{f}(t)\|=g(t)$, where $f:M\to \mathbb R^2$. We construct a probability measure on the space of almost everywhere differentiable solutions of this differential equation and study this measure. A solution of this equation can be viewed as an isometric immersion of a compact 1-manifold into $\mathbb R^2$. Nash's convergence technique in the proof of isometric $C^1$-immersion theorem plays an important role in the construction.