Equidimensional Isometric Extensions

TL;DR

This paper investigates the isometric extension problem for hypersurfaces, establishing curvature obstructions for differentiable and Lipschitz extensions, and constructs Lipschitz isometric extensions with controlled singularities, including collapsing mappings of spheres.

Contribution

It introduces new curvature obstructions for Lipschitz extensions and constructs one-sided Lipschitz isometric extensions using convex integration techniques.

Findings

Curvature obstruction for differentiable extensions

Obstruction to Lipschitz extensions via length comparison

Existence of Lipschitz isometries collapsing spheres

Abstract

Let $\Sigma$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:\Sigma\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differentiable extensions and an obstruction to the existence of Lipschitz extensions of $f$ using a length comparison argument. Using a weak form of convex integration, we then construct one-sided isometric Lipschitz extensions of which we compute the Hausdorff dimension of the singular set and obtain an accompanying density result. As an application we obtain the existence of infinitely many Lipschitz isometries collapsing the standard two-sphere to the closed standard unit $2$-disk mapping a great-circle to the boundary of the disk.