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The Weyl problem of isometric immersions revisited

Authors :
Siran Li
Publication Year :
2020
Publisher :
arXiv, 2020.

Abstract

We revisit the classical problem due to Weyl, as well as its generalisations, concerning the isometric immersions of $\mathbb{S}^2$ into simply-connected $3$-dimensional Riemannian manifolds with non-negative Gauss curvature. A sufficient condition is exhibited for the existence of global $C^{1,1}$-isometric immersions. Our developments are based on the framework \`{a} la Labourie (Immersions isom\'{e}triques elliptiques et courbes pseudo-holomorphes, J. Diff. Geom. 30 (1989), 395--424) of studying isometric immersions using $J$-holomorphic curves. We obtain along the way a generalisation of a classical theorem due to Heinz and Pogorelov.<br />Comment: 11 pages

Details

Database :
OpenAIRE
Accession number :
edsair.doi.dedup.....3bf61de47fdbf831bfcfe3d517088a5f
Full Text :
https://doi.org/10.48550/arxiv.2004.05532