Fourier dimension of constant rank hypersurfaces

2024/10/13 by J. Zhu, Zhu, Junjie
Mathematics · #Geometric Analysis and Curvature Flows #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2410.09711

Abstract

Any hypersurface in ℝ^d+1 has a Hausdorff dimension of d. However, the Fourier dimension depends on the finer geometric properties of the hypersurface. For example, the Fourier dimension of a hyperplane is 0, and the Fourier dimension of a hypersurface with non-vanishing Gaussian curvature is d. Recently, Harris showed that the Euclidean light cone in ℝ^d+1 has a Fourier dimension of d-1, which leads one to conjecture that the Fourier dimension of a hypersurface equals the number of non-vanishing principal curvatures. We prove this conjecture for all constant rank hypersurfaces. Our method involves substantial generalizations of Harris's strategy.

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