[Paper Review] Embedded constant mean curvature hypersurfaces on spheres

Óscar Perdomo|ArXiv.org|Mar 7, 2009

Geometric Analysis and Curvature Flows7 citations

TL;DR

This paper constructs non-isoparametric, compact, embedded constant mean curvature (CMC) hypersurfaces in $S^{n+1}$ for $n \geq 2$, $m \geq 2$, with symmetry group $O(n) \times \mathbb{Z}_m$, by analyzing profile curves invariant under $2\pi/m$ rotations. It proves that for $H$ between $\cot(\pi/m)$ and $b_{m,n} = \frac{(m^2-2)\sqrt{n-1}}{n\sqrt{m^2-1}}$, such embedded CMC hypersurfaces exist, and shows that every $H \neq 0, \pm 1/\sqrt{3}$ arises as the mean curvature of a non-isoparametric CMC surface in $S^3$. The results extend to hyperbolic and Euclidean spaces with analogous constructions.

ABSTRACT

Let m>1 and n>1 be any pair of integers. In this paper we prove that if H is between the numbers \cot(\fracπ{m}) and b_{m,n}=\frac{(m^2-2)\sqrt{n-1}}{n\sqrt{m^2-1}}, then, there exists a non isoparametric, compact embedded hypersurface in S^{n+1} with constant mean curvature H that admits the group O(n)x Z_m in their group of isometries, here O(n) is the set of n x n orthogonal matrices and Z_m are the integers mod m. When m=2 and H is close to the boundary value 0, the hypersurfaces look like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When m>2 and H is close to \cot(\fracπ{m}), the hypersurfaces look like a necklets made out of m spheres with (m+1) catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when H is close to b_{m,n} the hypersurface is close to an isoparametric hypersurface with the same mean curvature. As a consequence of the expression of these bounds for H, we have that every H different from 0,\pm\frac{1}{\sqrt{3}} can be realized as the mean curvature of a non isoparametric CMC surface in S^3. For hyperbolic spaces we prove that every non negative H can be realized as the mean curvature of an embedded CMC hypersurface in H^{n+1}, moreover we prove that when H>1 this hypersurface admits the group O(n) imes Z in its group of isometries. Here Z are the integer numbers. As a corollary of the properties proven for these hypersurfaces, for any n> 5, we construct non isoparametric compact minimal hypersurfaces in S^{n+1} which cone in R^{n+2} is stable. Also, we will prove that the stability index of every non isoparametric minimal hypersurface with two principal curvatures in S^{n+1} is greater than 2n+5.

Motivation & Objective

• To construct non-isoparametric, compact, embedded CMC hypersurfaces in $S^{n+1}$ with prescribed symmetry $O(n) \times \mathbb{Z}_m$.
• To determine the range of mean curvature $H$ for which such embedded hypersurfaces exist.
• To extend the construction to hyperbolic and Euclidean spaces and establish embeddedness and symmetry properties.
• To analyze the stability index of minimal hypersurfaces with two principal curvatures in $S^{n+1}$, showing it exceeds $2n+5$.
• To demonstrate that certain minimal hypersurfaces in $S^{n+1}$ cone to stable minimal cones in $\mathbb{R}^{n+2}$ for $n \geq 6$.

Proposed method

• Analyzes the profile curve of CMC hypersurfaces by separating radial and angular components, proving the angle function is strictly increasing to ensure injectivity and embedding.
• Uses the ODE $ (g')^2 = g^{2-2n} q(g) $ with $ q(v) = C v^{2n-2} - (H^2 - 1)v^{2n} - 2H v^n - 1 $ to model the profile curve in $S^{n+1}$.
• Applies a modified version of Otsuki’s method by tracking radius and angle separately, avoiding reliance on the supporting function.
• Defines the total rotation angle $ K = \int_0^T \frac{r(u)\lambda(u)}{1 + r^2(u)} du $ to determine periodicity and symmetry under $\mathbb{Z}_m$ rotations.
• Adapts the method to hyperbolic space using a modified ODE with $ H > 1 $, proving periodic solutions and embeddedness.
• Uses geometric integration and vector field analysis (e.g., $\eta$, $\nu$) to reconstruct the hypersurface from the profile curve and confirm constant mean curvature.

Experimental results

Research questions

• RQ1 For which values of $H$ does there exist a non-isoparametric, compact, embedded CMC hypersurface in $S^{n+1}$ with $O(n) \times \mathbb{Z}_m$ symmetry?
• RQ2 Can every $H \neq 0, \pm 1/\sqrt{3}$ be realized as the mean curvature of a non-isoparametric CMC surface in $S^3$?
• RQ3 What is the stability index of non-isoparametric minimal hypersurfaces with two principal curvatures in $S^{n+1}$?
• RQ4 Can embedded CMC hypersurfaces in $H^{n+1}$ be constructed with $H > 1$ and $O(n) \times \mathbb{Z}$ symmetry?
• RQ5 Do certain minimal hypersurfaces in $S^{n+1}$ cone to stable minimal cones in $\mathbb{R}^{n+2}$ for $n \geq 6$?

Key findings

• For $H \in \left(\cot\left(\frac{\pi}{m}\right), \frac{(m^2-2)\sqrt{n-1}}{n\sqrt{m^2-1}}\right)$, there exists a non-isoparametric, compact, embedded CMC hypersurface in $S^{n+1}$ with $O(n) \times \mathbb{Z}_m$ symmetry.
• When $m=2$ and $H \to 0^+$, the hypersurfaces resemble two nearly coinciding $n$-spheres connected by two catenoid necks.
• When $m>2$ and $H \to \cot(\pi/m)^+$, the hypersurfaces resemble a necklace of $m$ spheres connected by $m+1$ catenoid necks.
• For $n \geq 6$, the paper constructs non-isoparametric compact minimal hypersurfaces in $S^{n+1}$ whose cone in $\mathbb{R}^{n+2}$ is stable.
• The stability index of every non-isoparametric minimal hypersurface with two principal curvatures in $S^{n+1}$ is greater than $2n+5$.
• In $H^{n+1}$, every $H \geq 0$ can be realized as the mean curvature of an embedded CMC hypersurface, and for $H > 1$, such hypersurfaces admit $O(n) \times \mathbb{Z}$ symmetry.