2020/10/17 by Jiaying Cai, Cai, Jiaying
Mathematics · #FOS: Mathematics #General Mathematics (math.GM) #Geometric Analysis and Curvature Flows #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2010.10259
openalex publication_date 2020/10/17 · openalex created_date 2020/10/29 · openalex updated_date 2026/07/28
Carathéodory's well-known conjecture states that every sufficiently smooth, closed convex surface in three dimensional Euclidean space admits at least two umbilic points. It has been established that the conjecture is true for all rotationally symmetric surfaces; in this paper, we investigate the umbilic points of two families of surfaces without rotational symmetry, and compute their indices. In particular, we find that the family of surfaces of the form ax2k+by2k+cz2k=1 with a,b,c>0, k∈ℤ>1 admit 14 umbilic points: six of one known form and eight of another. For many tested values of a,b,c,k, such umbilic points have indices -1/2 and 1, respectively. We also explore the dependence of the umbilic points on the parameter ε of the surface ax2+εx4+ay2+εy4+bz2=1. In particular, for both a<b>b, there exist exactly two umbilic points with index 1 for ε smaller than certain critical values. For larger ε, surfaces with a>b admit exactly ten umbilic points; for many tested values of a,b,ε, these points have indices 1/2 and -1. For larger ε, surfaces with a<b></b></b>