JournalszaaVol. 40, No. 3pp. 349–366

Equidimensional Isometric Extensions

  • Micha Wasem

    HTA Freiburg, Switzerland
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Let Σ\Sigma be a hypersurface in an nn-dimensional Riemannian manifold MM, n2n\geq 2. We study the isometric extension problem for isometric immersions f ⁣:ΣRnf\colon\Sigma\to\mathbb{R}^n, where Rn\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 ff 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 22-disk mapping a great-circle to the boundary of the disk.

Cite this article

Micha Wasem, Equidimensional Isometric Extensions. Z. Anal. Anwend. 40 (2021), no. 3, pp. 349–366

DOI 10.4171/ZAA/1688