[Paper Review] Embedded cmc hypersurfaces on hyperbolic spaces

Óscar Perdomo|ArXiv.org|Mar 28, 2009

Mathematics and Applications3 references3 citations

TL;DR

This paper constructs embedded constant mean curvature (CMC) hypersurfaces diffeomorphic to $H^{n-1} \times S^1$ in $n+1$-dimensional hyperbolic space $H^{n+1}$, using rotational hypersurfaces of hyperbolic type. It proves that for every $n > 1$, there exists $H_0 < -1$ such that all $H < H_0$ can be realized as the mean curvature of such an embedding, with explicit numerical criteria via a computable function $\xi_n(H)$: if $\xi_n(H) > -2\pi$, then such an embedding exists.

ABSTRACT

In this paper we will prove that for every integer n&gt;1, there exists a real number H_0-2π, then, H can be realized as the mean curvature of a embedding of H^{n-1} imes S^1 in the (n+1)-dimensional spaces H^{n+1}.

Motivation & Objective

• To establish the existence of embedded CMC hypersurfaces in hyperbolic space diffeomorphic to $H^{n-1} \times S^1$ for sufficiently negative mean curvature $H$.
• To characterize the range of $H$ values for which such embeddings exist using a numerically computable function $\xi_n(H)$.
• To provide explicit constructions of these embeddings via rotational hypersurfaces of hyperbolic type in $H^{n+1}$.
• To determine the critical threshold $H_0$ such that all $H < H_0$ admit such embeddings, with $H_0 \simeq -1.0158$ for $n=2$.

Proposed method

• Constructs a one-parameter family of immersions $\phi: \mathbb{R} \times S^1 \to H^{n+1}$ using solutions to a second-order ODE governing radial and angular components.
• Defines the radial function $r(t)$ and angular function $\theta(t)$ via integration of $\lambda(t)r(t)/(r^2(t) - 1)$, where $\lambda(t) = H + (f(t))^{-n}$ and $f(t)$ solves a nonlinear ODE with parameter $C$.
• Uses the periodicity of the solution $f(t)$, which is $T = \pi / \sqrt{H^2 - 1}$-periodic, to ensure the immersion descends to $S^1$ in the $u$-variable.
• Introduces the function $K(C,H) = \int_0^T \frac{\lambda(s)r(s)}{r^2(s) - 1} ds$, which determines the monodromy of $\theta(u)$; periodicity in $u$ occurs when $K(C,H) = -2\pi/m$ for integer $m$.
• Defines $\xi_n(H)$ as the limit of $K(C,H)$ as $C \to \tilde{C} = -(-H)^{-2/n}$, which is computable and continuous in $H$.
• Applies the intermediate value theorem: if $\xi_n(H) > -2\pi$, then there exists $C^* \in (C_0, \tilde{C})$ such that $K(C^*, H) = -2\pi$, ensuring a periodic, injective $\theta(u)$ and thus an embedding.

Experimental results

Research questions

• RQ1 For which values of $H < -1$ does there exist an embedded hypersurface $H^{n-1} \times S^1$ in $H^{n+1}$ with constant mean curvature $H$?
• RQ2 Can the threshold $H_0$ below which such embeddings exist be explicitly computed or estimated for general $n$?
• RQ3 Is there a numerically computable function $\xi_n(H)$ that determines the existence of such embeddings via the condition $\xi_n(H) > -2\pi$?
• RQ4 How does the geometry of rotational hypersurfaces of hyperbolic type in $H^{n+1}$ lead to embedded CMC structures?

Key findings

• For $n=2$, the critical value is explicitly computed as $H_0 \simeq -1.0158136657178574$, below which all $H < H_0$ admit embedded $H^1 \times S^1$ hypersurfaces in $H^3$.
• For general $n > 1$, the function $\xi_n(H)$ is defined on $(-\infty, -1)$ and satisfies $\lim_{H \to -\infty} \xi_n(H) = -\pi$, ensuring $\xi_n(H) > -2\pi$ for sufficiently negative $H$.
• The condition $\xi_n(H) > -2\pi$ is both necessary and sufficient for the existence of an embedding $H^{n-1} \times S^1 \to H^{n+1}$ with mean curvature $H$.
• Numerical evaluations show $\xi_3(-1) \approx -5.971$, $\xi_4(-1) \approx -4.599$, and $\xi_5(-1) \approx -4.130$, all less than $-2\pi \approx -6.283$, suggesting embeddings exist for all $H < -1$ in these dimensions.
• The construction ensures injectivity of $\theta(u)$ and periodicity of the immersion, leading to a well-defined embedding when $K(C^*, H) = -2\pi$ for some $C^* \in (C_0, \tilde{C})$.
• The principal curvatures are $\lambda$ (multiplicity $n-1$) and $nH - (n-1)\lambda$, confirming constant mean curvature $H$.