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

Abstract

Let $M$ be a compact 1-manifold. Given a continuous function $g:M\to {\mathbb{R}}_{+}$ we consider the following ordinary differential equation: $\Vert \dot{f}(t)\Vert =g(t)$, 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.