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arXiv · 1509.08584

Isometric Immersion of Surface with Negative Gauss Curvature and the Lax-Friedrichs Scheme

Wentao Cao·Feimin Huang·Dehua Wang
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Abstract

The isometric immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large $L^\infty$ solution is obtained which leads to a $C^{1,1}$ isometric immersion. The approximate solutions are constructed by the the Lax-Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The $H^{-1}$ compactness is also derived. A compensated compactness framework is applied to obtain the existence of large $L^\infty$ solution to the Gauss-Codazzi equations for the surfaces more general than those in literature.

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Wentao Cao, Feimin Huang, Dehua Wang. 2015-12-19. Isometric Immersion of Surface with Negative Gauss Curvature and the Lax-Friedrichs Scheme. https://arxiv.org/abs/1509.08584

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