Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part II

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 it is non-embedded ("nodoid type"). As b increases toward B, the hypersurfaces converge to a minimal hypersurface with two singular points.