[Paper Review] Constant mean curvature surfaces in $\mathbb{H}^2 imes\mathbb{R}$ with boundary in two parallel planes

Rodrigo B. Soares, Patrícia Kruse Klaser+1 more|arXiv (Cornell University)|Dec 23, 2015

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper establishes existence and non-existence conditions for constant mean curvature (CMC) surfaces in $ H^2 \times \mathbb{R}$ with boundaries in two parallel planes. Using geometric barriers—H-nodoids, catenoids, and H-spherical caps—it proves that for $H \geq 0$, a unique solution exists when the distance between planes and boundary curves satisfy specific curvature and height constraints, with special behavior for $H > 1/2$.

ABSTRACT

Given $H\in [0,\infty),$ some sufficient conditions for existence of CMC $H$ graphs with boundary in two parallel planes of $\mathbb{H}^2 imes\mathbb{R}$ are presented. Height estimates for outwards-oriented CMC surfaces (horo)cyllindrically bounded are also exhibited.

Motivation & Objective

• To establish sufficient conditions for the existence of CMC $H$ graphs with boundary in two parallel planes in the product space $ H^2 \times \mathbb{R}$.
• To analyze the behavior of CMC surfaces when $H > 1/2$, a regime not previously treated in the literature.
• To provide height estimates and non-existence results for CMC surfaces that are horo-cylindrically bounded.
• To construct geometric barriers—H-nodoids, catenoids, and H-spherical caps—based on rotational graphs of specific functions in $ H^2$.

Proposed method

• Constructs the H-nodoid function ${\rm H\text{-}nod}_r(s)$ as a rotationally symmetric solution to the CMC equation in $ H^2$, defined via an integral involving hyperbolic functions and $H$.
• Uses the H-nodoid as a barrier, with its domain restricted to $[0, T_H]$, where $T_H = \infty$ for $H \leq 1/2$ and $T_H = \ln\left(\frac{2H+1}{2H-1}\right)$ for $H > 1/2$, independent of the neck radius $r$.
• Applies Perron’s method and the method of sub- and supersolutions using barriers derived from H-nodoids, catenoids ($H=0$ case), and H-spherical caps.
• Imposes boundary conditions via the Dirichlet problem $Q_H(u) = 0$ in $ H^2$, with $u = h$ on $ hd$ and $u = 0$ on $ hd$, where $ hd$ is the annular domain bounded by the projections of the boundary curves.
• Employs the Tangency Principle to prove non-existence: if a CMC surface lies in a slab of height $h > h_H$, it must coincide with a horospherical surface, leading to contradiction.
• Uses the H-horocylinder foliation and reflection symmetry of the H-nodoid to derive non-existence for $h > 2\max{\rm H\text{-}nod}_{r^*}$ when $r < r^*$.

Experimental results

Research questions

• RQ1 Under what conditions does a CMC $H$ graph exist with boundary in two parallel planes in $ H^2 \times \mathbb{R}$ for $H \geq 0$?
• RQ2 How does the behavior of CMC barriers—specifically H-nodoids—change when $H > 1/2$, and what implications does this have for existence?
• RQ3 What height estimates can be derived for CMC surfaces that are hore-cylindrically bounded in $ H^2 \times \mathbb{R}$?
• RQ4 Can non-existence results be established for CMC surfaces in slabs of $ H^2 \times \mathbb{R}$ based on the mean curvature $H$ and slab height $h$?
• RQ5 How do the interior and exterior circle conditions on the boundary curves $ hd$ and $ hd$ influence the solvability of the Dirichlet problem for CMC graphs?

Key findings

• For $H \leq 1/2$, existence holds if the distance $d$ between the boundary curves satisfies $\cosh(r + d/2) \leq \cosh(r) + \frac{\sinh(r)}{2H}$, and $h \leq \max\left\{ \text{cat}_r(d), \frac{2Hd}{\sqrt{\coth^2(r) - 4H^2}} \right\}$, with the latter term finite only if $\coth^2(r) > 4H^2$.
• For $H > 1/2$, existence requires $R \leq T_H = \ln\left(\frac{2H+1}{2H-1}\right)$ and $h \leq \frac{4H}{\sqrt{4H^2 - 1}} \arctan\left(\sqrt{\frac{1 - 4H^2 \tanh^2\left(\frac{T_H - d}{2}\right)}{4H^2 - 1}}\right)$, with $T_H$ independent of the neck radius $r$.
• The H-nodoid's domain length $T_H$ is independent of the inner radius $r$, implying that the distance between vertical slope points of the H-nodoid does not scale with the neck size.
• The H-spherical cap is recovered as the limit $r \to 0$ of the H-nodoid, and its explicit elementary form allows it to be used implicitly in existence criteria.
• Non-existence is proven for $h > h_H$, where $h_H$ is the height of the H-horocylinder, using a foliation of $B \times [0, h_H]$ by H-horospherical surfaces and the Tangency Principle.
• For $H > 1/2$, non-existence holds if $h > 2\max{\rm H\text{-}nod}_{r^*}$, provided a suitable H-nodoid of radius $r < r^*$ exists and touches the surface $M$ from inside.