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arXiv · 2606.16766

Restriction estimates for surfaces with negative curvature in $\mathbb R^3$

Shaoming Guo·Diankun Liu·Yakun Xi
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Abstract

In $\mathbb R^3$, we prove that $L^q\to L^p$ restriction estimates associated with smooth surfaces with negative Gaussian curvature hold for all $p>\frac{22}{7}$ and $q'<\frac{p}{2}$. Building on Demeter--Wu's work for the model hyperbolic paraboloid, we introduce a geometric propagation principle for good lines, which controls the degenerate directions arising from straight-line segments on the surface. This overcomes a key difficulty in the general case, where such directions may vary with the geometry rather than being fixed by the coordinate axes.

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Shaoming Guo, Diankun Liu, Yakun Xi. 2026-06-15. Restriction estimates for surfaces with negative curvature in $\mathbb R^3$. https://arxiv.org/abs/2606.16766

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