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

TL;DR

The paper constructs a new one-parameter family of constant mean curvature hypersurfaces in $oldsymbol{S}^{4}$ by desingularizing a carefully designed piecewise-CMC generating surface that glues two totally umbilical 3-spheres to two Clifford pieces along four circles. The CMC condition reduces to an autonomous ODE for a profile curve, enabling numerical continuation along a curve of triples $(a,H,T)$; this yields a sequence of embedded and non-embedded examples, with a minimal singular limit at $H=0$ and convergence to a second singular surface $M_f$ that is minimal except at two poles. The result generalizes the Delaunay-type desingularization from Euclidean space to the sphere, enriching the family of compact CMC hypersurfaces and illustrating how singular limits can seed families with diverse topology. The findings also demonstrate a clear transition from unduloid-type embeddings to nodoid-type non-embeddings within a sphere setting, and provide explicit numerical data detailing the evolution of mean curvature along the family. Overall, the work advances understanding of generalized rotational CMC hypersurfaces in spheres and highlights a robust numerical framework for exploring such geometric families.

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.

Figures (13)

• Figure 1: The piecewise–CMC hypersurface defined in Equation \ref{['mM']} corresponds to a periodic solution $(\theta(t),f_{1}(t),f_{2}(t))$ that exhibits four singularities. The graphs of $f_{1}$ and $f_{2}$ are shown on the left, and the graph of $\theta$ is shown on the right.
• Figure 2: The CMC hypersurface $M_f$ defined in Equaation \ref{['fig:gfMf']} can be regarded as a periodic solution of the $(\theta(t),f_1(t), f_2(t)$ with three singularities. Here we show the graphs of $f_1$ and $f_2$ associated with this solution with singularities
• Figure 3: Graph of the curve $\Lambda$ that solve the system of equations \ref{['theaHTsytem']}. Starting form the limit point $Z_0$ (the one with H$H<0$, we notice that the values of $H$ start decreasing reaching a minimum value of $H$, then, they increase until they reach a positive maximum and finally $H$ decreases to zero to approach the limit point $Z_f$. We also show the projection of $\Lambda$ on the $a-H$ plane.
• Figure 4: Graph of the profile curve associated with the solution $Z_1=(a,H,T)=(0.577096, -0.707791, 2.30054)$. This solution provides a smooth embedded CMC with curvature $\mathbf{S}^{4}$ very close to the manifold $M$ defined in Equation \ref{['mM']}.
• Figure 5: Graph of the profile curve associated with the solution $Z_2=(a,H,T)=(0.514328, -0.844304, 2.26274)$. This solution provides an embedded CMC hypesurface in$\mathbf{S}^{4}$

Theorems & Definitions (13)

• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Theorem 3.5
• proof
• Remark 3.6