A Liouville-type theorem for cylindrical cones

2023/01/14 by Nick Edelen, Edelen, Nick, Gábor Székelyhidi +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2301.05967

openalex publication_date 2023/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that C₀ⁿ ⊂ ℝⁿ⁺¹ is a smooth strictly minimizing and strictly stable minimal hypercone, l ≥ 0, and M a complete embedded minimal hypersurface of ℝ^n+1+l lying to one side of C = C₀ × ℝ^l. If the density at infinity of M is less than twice the density of C, then we show that M = H(λ) × ℝ^l, where \H(λ)\_λ is the Hardt-Simon foliation of C₀. This extends a result of L. Simon, where an additional smallness assumption is required for the normal vector of M.

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