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

Oscar Perdomo

TL;DR

This work constructs and analyzes a new, two-parameter family of embedded CMC hypersurfaces in spheres $S^{n+1}$ by augmenting rotational symmetry with an additional $S^l$ factor and a pair of periodic profile functions. The mean curvature condition reduces to a system of ODEs for the profile, and numerical methods yield closed, embedded profiles corresponding to $S^{n-\ell-2}\times S^l\times S^1$ topologies; the resulting $H$-parameter curve exhibits rich behavior including a negative minimum, a minimal example at $H=0$, and two distinct embeddings for most negative $H$. The paper also provides a quantitative volume comparison between these new embedded examples and Clifford hypersurfaces, supplying explicit values up to $n=12$ and demonstrating that Clifford hypersurfaces often have smaller volume than the non-Clifford members of the new family. In the $S^4$ case, the authors obtain a rigorous local existence result near small $|H|$ via analytic continuation from a known hypertorus, connecting to Carlotto–Schulz’s minimal example. Collectively, the results broaden the catalog of compact embedded CMC hypersurfaces in spheres, offer a practical numerical framework for constructing and validating such examples, and furnish evidence supporting Yau’s conjecture in this extended setting.

Abstract

In this paper, we describe a family of embedded hypersurfaces with constant mean curvature (CMC) in the $(n+1)$-dimensional unit sphere. In the process, we provide evidence for new CMC embedded examples. In particular, for some examples with $H=0$, we verify Yau's conjecture stating that among the embedded, non-totally umbilical minimal hypersurfaces in spheres, the Clifford hypersurfaces have the least area.

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

TL;DR

This work constructs and analyzes a new, two-parameter family of embedded CMC hypersurfaces in spheres by augmenting rotational symmetry with an additional factor and a pair of periodic profile functions. The mean curvature condition reduces to a system of ODEs for the profile, and numerical methods yield closed, embedded profiles corresponding to topologies; the resulting -parameter curve exhibits rich behavior including a negative minimum, a minimal example at , and two distinct embeddings for most negative . The paper also provides a quantitative volume comparison between these new embedded examples and Clifford hypersurfaces, supplying explicit values up to and demonstrating that Clifford hypersurfaces often have smaller volume than the non-Clifford members of the new family. In the case, the authors obtain a rigorous local existence result near small via analytic continuation from a known hypertorus, connecting to Carlotto–Schulz’s minimal example. Collectively, the results broaden the catalog of compact embedded CMC hypersurfaces in spheres, offer a practical numerical framework for constructing and validating such examples, and furnish evidence supporting Yau’s conjecture in this extended setting.

Abstract

In this paper, we describe a family of embedded hypersurfaces with constant mean curvature (CMC) in the -dimensional unit sphere. In the process, we provide evidence for new CMC embedded examples. In particular, for some examples with , we verify Yau's conjecture stating that among the embedded, non-totally umbilical minimal hypersurfaces in spheres, the Clifford hypersurfaces have the least area.

Paper Structure

This paper contains 22 sections, 4 theorems, 70 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Totally Umbilical Hypersurfaces
  3. Clifford hypersurfaces
  4. Rotational hypersurfaces
  5. The new family of CMC in the sphere
  6. The ODE
  7. Description of the new CMC hypersurfaces in the sphere.
  8. Numerical construction of the embedded CMC hypersurfaces
  9. CMC Hypersurfaces in $S^4$
  10. Describing the profile curve for the generalize rotational embedded CMC in $S^4$
  11. The Volume some of the minimal examples
  12. Case $n=3$
  13. Case $n=4$
  14. Case $n=5$
  15. Case $n=6$
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $k$ and $l$ be positive integers such that $n = k + l + 1$. If $f_1$ and $f_2$ are $T$-periodic functions satisfying $f_2 > 0$ and $f_1^2 + f_2^2 < 1$, and they satisfy the system where then the immersion defines a hypersurface with constant mean curvature $H$.

Figures (7)

  • Figure 1: Each point on the curve $r = \frac{1}{\sqrt{1+H^2}}$ represents a CMC hypersurface. Specifically, the point $\left(H, \frac{1}{\sqrt{1+H^2}}\right)$ corresponds to the hypersurface given by the immersion (\ref{['imtu']}) with $r = \frac{1}{\sqrt{1+H^2}}$.
  • Figure 2: Graphs of the functions $f_1$, $f_2$, and $\theta$ associated with the point $q_0$. Notice that there are two highlighted points in the left graph and one in the right graph. These indicate that $f_2(0) = 0.187605...$, which is the first entry of $q_0$, and that $f_1(1.15925...) = 0$ and $\theta(1.15925...) = \pi$.
  • Figure 3: Profile curve of a minimal hypersurface in $S^4$.
  • Figure 4: On the left, we depict the curve $\Gamma$, consisting of points that solve system \ref{['system1']} for $n=3$. On the right, we show the projection of $\Gamma$ onto the $T$-$H$ plane.
  • Figure 5: Profile curves with $a\in (a_{31}^{H_{31}^{\min}},a^{H=0}_{31})$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Definition 3.1