[Paper Review] Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space

Shoichi Fujimori, Yu Kawakami+4 more|Osaka City University (Osaka City University)|Sep 19, 2015

Geometric Analysis and Curvature Flows1 references3 citations

TL;DR

This paper establishes that Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space admit a canonical analytic extension to properly embedded zero mean curvature surfaces of mixed type, transitioning from space-like to time-like regions. The extension is shown to be injective and proper, preserving the rotational symmetry and embedding structure of the original surface.

ABSTRACT

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2π/n)$-rotation symmetry with respect to its axis. In this paper, we show that the corresponding maximal surface $f_n$ in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ has an analytic extension $ ilde f_n$ as a properly embedded zero mean curvature surface. The extension changes type into a time-like (minimal) surface.

Motivation & Objective

• To investigate the analytic extension properties of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space.
• To determine whether the singular, space-like maximal surfaces with fold singularities admit a smooth, globally defined extension to time-like minimal surfaces.
• To establish the embeddedness and properness of the extended surface in the Lorentz-Minkowski 3-space.
• To analyze the geometric and topological structure of the analytic extension using Weierstrass representation and symmetry arguments.
• To prove that the extended surface is a proper embedding, even across the transition from space-like to time-like regions.

Proposed method

• Utilizes the Weierstrass representation for maximal surfaces in Lorentz-Minkowski 3-space using meromorphic data $(g_n, ω_n) = (z^{n-1}, \frac{i\,dz}{(z^n - 1)^2})$.
• Applies the holomorphic lift $F = \int (-2g, 1+g^2, i(1-g^2))\omega$ to construct the maximal surface $f_n$ as the real part of $F$.
• Employs rotational symmetry of order $n$ to reduce the analysis to a fundamental domain and study contour lines of constant $x_0$-coordinate.
• Analyzes the behavior of the extension map $\tilde{f}_n$ on the universal cover $\Omega_n$ by examining the image of contour lines $\tilde{x}_0^{-1}(h)$.
• Uses Chebyshev polynomials $U_{n-1}(x)$ and $T_n(x)$ to analyze monotonicity and positivity of key functions $\Upsilon(u)$ and $\Phi(h)$.
• Proves injectivity of $\tilde{f}_n$ by showing that the image of each contour line does not intersect its rotated copies under the $n$-fold symmetry.

Experimental results

Research questions

• RQ1 Can the singular, space-like Jorge-Meeks type maximal surface in Lorentz-Minkowski 3-space be analytically extended to a globally defined, smooth surface of mixed type?
• RQ2 Does the analytic extension of the Jorge-Meeks $n$-noid maximal surface remain embedded and properly immersed in $\mathbb{R}^3_1$?
• RQ3 How does the rotational symmetry of order $n$ influence the injectivity and global structure of the analytic extension?
• RQ4 What role do Chebyshev polynomials play in proving the positivity and monotonicity of the extension's Jacobian or energy functional?
• RQ5 Is the extended surface a proper embedding, and does it avoid self-intersections across the transition from space-like to time-like regions?

Key findings

• The analytic extension $\tilde{f}_n$ of the Jorge-Meeks type maximal surface $f_n$ is a properly embedded zero mean curvature surface in $\mathbb{R}^3_1$.
• The extension is injective and proper, as shown by proving that the image of each contour line $\tilde{x}_0^{-1}(h)$ does not intersect its rotated copies under the $n$-fold symmetry.
• For $n=2$, the extension is trivially a proper embedding; for $n \geq 3$, injectivity is established via analysis of the function $\Phi(h)$ and the positivity of $\Upsilon(u)$.
• The function $\Upsilon(u)$, derived from Chebyshev polynomials, is strictly positive for all $u > \cos\theta_0$, ensuring $\frac{d\Phi}{dh} > 0$ and thus $\Phi(h) > 0$ for all $h > 0$.
• The range of $U_{n-1}(x)$ on $[\cos(\pi/n), \infty)$ is $[-1, \infty)$, and $U_{n-1}(\cos(\pi/n)) = 0$, which supports the positivity of key expressions in the injectivity proof.
• The extended surface $\tilde{f}_n$ changes type from space-like to time-like across the unit circle $|z| = 1$, with the time-like parts clearly visible in the figures as black-shaded regions.