The One-Sided Isometric Extension Problem

Abstract

Let $\Sigma$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n⩾2$. We give a necessary condition for an isometric immersion of $\Sigma$ into ${\mathbb{R}}^q$ equipped with the standard Euclidean metric, $q⩾n+1$, to be locally isometrically $C^1$-extendable to $M$. Even if this condition is not met, "one-sided" isometric $C^1$-extensions may exist and turn out to satisfy a $C^0$-dense parametric $h$-principle in the sense of Gromov.

Final Version, to appear in Results in Mathematics, 33 pages, 5 figures

References in corpus (1)

• The $h$-principle and the equations of fluid dynamics

Cited by in corpus (3)

• Global Nash-Kuiper theorem for compact manifolds
• Equidimensional Isometric Extensions
• Very weak solutions to the two dimensional Monge-Ampére equation