2025/05/14 by Óscar Perdomo, Perdomo, Oscar · 1 citation
Computer Science · Engineering · Mathematics · #53A10 (Primary) #53C21 #53C42 #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2505.09102
openalex publication_date 2025/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In R3, let M be the infinite union of unit spheres whose centers lie at even integers on the x-axis; every pair of consecutive spheres touches at (2m+1, 0, 0). Desingularizing these point contacts yields Delaunay's classical constant mean curvature (CMC) surfaces, including unduloids and nodoids. Motivated by this picture, we construct an analogue in the unit sphere S4. We begin with the piecewise-smooth hypersurface M contained in S4, obtained by gluing two carefully chosen totally umbilical 3-spheres to two specific Clifford hypersurfaces, all four components sharing the same constant mean curvature and meeting along four disjoint circles. We provide numerical evidence that these circles can be desingularized: there exists a smooth one-parameter family Sigmab, each lying in S4, of CMC hypersurfaces such that Sigmab approaches M as b tends to 0. The mean curvature H(b) varies smoothly along the family and vanishes at a single non-embedded minimal member. Moreover, there is a threshold B1 in (0, B) such that when b < B1 the hypersurface Sigmab is embedded ("unduloid type"), whereas for b >= B1 it is non-embedded ("nodoid type"). As b increases toward B, the hypersurfaces converge to a minimal hypersurface with two singular points.