Calculus of Variations and Geometric Measure Theory
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C. De Lellis - D. Inauen

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

created by delellis on 18 Sep 2018
modified on 13 Feb 2019



Inserted: 18 sep 2018
Last Updated: 13 feb 2019

Year: 2018


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.


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