Calculus of Variations and Geometric Measure Theory

C. De Lellis - D. Inauen

$C^{1,\alpha} isometric embeddings of polar caps

created by delellis on 18 Sep 2018
modified on 01 Dec 2020


Published Paper

Inserted: 18 sep 2018
Last Updated: 1 dec 2020

Journal: Adv. Math.
Volume: 363
Year: 2020


We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the H\"older space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $\alpha > \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any $\alpha < \frac{1}{2}$ we construct $C^{1,\alpha}$ isometric embeddings of portions of the standard $2$-dimensional sphere for which such property fails.