Abstract
We address the problem of global embedding of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. A theorem of Efimov states that if the curvature decays too slowly to zero then global smooth immersion is impossible. On the other hand a theorem of J.-X. Hong shows that if decay is sufficiently rapid (roughly like t −(2+δ) for δ > 0) then global smooth immersion can be accomplished. Here we present recent results on applying the method of compensated compactness to achieve a non-smooth global immersion with rough data and we give an emphasis on the role of decay rate of the Gauss curvature.
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Christoforou, C., Slemrod, M. On the decay rate of the Gauss curvature for isometric immersions. Bull Braz Math Soc, New Series 47, 255–265 (2016). https://doi.org/10.1007/s00574-016-0136-z
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DOI: https://doi.org/10.1007/s00574-016-0136-z
Keywords
- isometric immersion problem
- Gauss curvature
- Gauss-Codazzi system
- systems of balance laws
- weak solutions
- compensated compactness