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

TL;DR

This paper constructs and numerically investigates a family of constant mean curvature hypersurfaces in S^4, generalizing classical Delaunay surfaces, and explores their embeddedness and singularity formation.

Contribution

It introduces a novel construction of CMC hypersurfaces in S^4 inspired by desingularization of sphere unions, and provides numerical evidence of their properties and bifurcations.

Findings

Existence of a smooth family of CMC hypersurfaces approaching a singular union of spheres.

Identification of a threshold parameter separating embedded and non-embedded hypersurfaces.

Convergence of hypersurfaces to a minimal hypersurface with singular points as the parameter varies.

Abstract

In R^3, 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 S^4. We begin with the piecewise-smooth hypersurface M contained in S^4, 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 Sigma_b, each lying in S^4, of CMC hypersurfaces such that Sigma_b 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 B_1 in (0, B) such that when b < B_1 the hypersurface Sigma_b is embedded ("unduloid type"), whereas for b >= B_1 it is non-embedded ("nodoid type"). As b increases toward B, the hypersurfaces converge to a minimal hypersurface with two singular points.