[Paper Review] The Alexandrov problem in a quotient space of $\mathbb H^2 imes \mathbb R$

Ana Menezes|arXiv (Cornell University)|Nov 14, 2011

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper establishes an Alexandrov-type theorem for compact embedded constant mean curvature surfaces in a quotient space of $ℝ\times\mathbb{R}$, proving that such surfaces must be rotational spheres. It further constructs periodic minimal surfaces and proves a multi-valued Rado-type theorem, showing that small perturbations of helicoidal boundaries in $ℝ\times\mathbb{R}$ yield unique minimal disks, extending classical results to this geometric setting.

ABSTRACT

We prove an Alexandrov type theorem for a quotient space of $\mathbb H^2 imes \mathbb R$. More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of $\mathbb H^2 imes \mathbb R$ by a subgroup of isometries generated by a parabolic translation along horocycles of $\mathbb H^2$ and a vertical translation. Moreover, we construct some examples of periodic minimal surfaces in $\mathbb H^2 imes\mathbb R$ and we prove a multi-valued Rado theorem for small perturbations of the helicoid in $\mathbb H^2 imes\mathbb R$.

Motivation & Objective

• To classify compact embedded constant mean curvature surfaces in the quotient space $ℝ\times\mathbb{R}/[ψ,T(h)]$, which is diffeomorphic to $ℝ^2\times\mathbb{R}$.
• To extend the Alexandrov reflection method to this non-simply connected quotient space with intrinsic flat tori of constant mean curvature.
• To construct explicit examples of doubly periodic minimal surfaces in $ℝ\times\mathbb{R}$ invariant under discrete isometry groups.
• To generalize Rado's classical uniqueness theorem for minimal graphs to minimal disks in $ℝ\times\mathbb{R}$ with boundaries that are small perturbations of helicoidal curves.

Proposed method

• Utilizes the Alexandrov reflection method adapted to the quotient space $ℝ\times\mathbb{R}/[ψ,T(h)]$, where $ψ$ is a horizontal translation along horocycles and $T(h)$ a vertical translation.
• Employs rotational symmetry and barrier methods via isometric reflections across horizontal geodesics to control boundary behavior and curvature estimates.
• Applies the maximum principle and boundary maximum principle to rule out non-trivial foliations or singularities in the intersection of minimal surfaces with reflection planes.
• Constructs minimal surfaces via a one-parameter family of Jordan curves $Γ_t$ converging to a perturbed helicoidal boundary, using uniform curvature estimates and convergence theorems.
• Uses rotation by $π$ around the $z$-axis to extend minimal disks to larger surfaces, enabling barrier constructions and uniqueness proofs.
• Applies the theory of multi-valued graphs and asymptotic boundaries to analyze the limit of minimal surfaces with perturbed helicoidal boundaries.

Experimental results

Research questions

• RQ1 Are compact embedded constant mean curvature surfaces in the quotient space $ℝ\times\mathbb{R}/[ψ,T(h)]$ necessarily rotational spheres?
• RQ2 Can periodic minimal surfaces be constructed in $ℝ\times\mathbb{R}$ that are invariant under a $K^2$-action generated by horocyclic and vertical translations?
• RQ3 Does a Rado-type uniqueness theorem hold for minimal disks in $ℝ\times\mathbb{R}$ whose boundaries are small perturbations of the boundary of a helicoid?
• RQ4 What is the behavior of minimal surfaces near the boundary when the asymptotic boundary is a small perturbation of a helicoid’s asymptotic boundary?
• RQ5 How do curvature estimates and reflection symmetries constrain the structure of minimal surfaces in this non-simply connected quotient space?

Key findings

• All compact embedded constant mean curvature surfaces in the quotient space $ℝ\times\mathbb{R}/[ψ,T(h)]$ are rotational spheres, establishing an Alexandrov-type theorem in this setting.
• The quotient space $ℝ\times\mathbb{R}/[ψ,T(h)]$ is diffeomorphic to $ℝ^2\times\mathbb{R}$ and admits a foliation by flat tori of constant mean curvature $1/2$, which are intrinsic to the geometry.
• For any small perturbation of a helicoidal boundary in $ℝ\times\mathbb{R}$, the unique minimal disk solving the Plateau problem is graphical and stable, extending Rado’s theorem to this setting.
• A sequence of minimal disks with boundaries converging to a perturbed helicoidal curve converges to a complete minimal surface with asymptotic boundary a small perturbation of the helicoid’s asymptotic boundary.
• The reflection principle via $π$-rotation around the $z$-axis allows extension of minimal surfaces across horizontal geodesics, enabling barrier constructions and uniqueness proofs.
• The proof of uniqueness for the perturbed helicoid boundary relies on contradiction: gluing two copies of the minimal disk leads to a sphere with singular foliations of negative index, which is impossible.