New free boundary minimal annuli of revolution in the 3-sphere

TL;DR

This work proves the existence of numerous free boundary minimal annuli of revolution in geodesic spheres within $\mathbb{S}^3$ by leveraging the do Carmo–Dajczer one-parameter family, including the Otsuki tori. It provides explicit constructions of complete minimal surfaces $X_a$ and analyzes when generating curves close to closed tori yield embedded annuli, uncovering an infinite, nested family of free boundary annuli with boundaries on geodesic spheres and, for suitable parameters, embedded examples contained in balls around radius $\pi/2$ (and sometimes larger). The authors then prove that each Otsuki torus hosts many immersed free boundary annuli, with precise lower bounds depending on the rational parameter $p/q$, and show that these surfaces enable extensions of recent isoperimetric and index bounds to geodesic balls of radius at least a hemisphere. Collectively, the results expand the catalog of explicit free boundary minimal surfaces in curved ambient spaces and provide tools for isoperimetric and index theory in geodesic balls beyond the hemisphere.

Abstract

We rigorously establish the existence of many free boundary minimal annuli with boundary in a geodesic sphere of $\mathbb{S}^3$. These arise as compact subdomains of a one-parameter family of complete minimal immersions of $\mathbb{R} \times \mathbb{S}^1$ into $\mathbb{S}^3$ described by do Carmo and Dajczer. While the immersed free boundary minimal annuli we exhibit may in general fail to be embedded or contained in a geodesic ball, we show that there is at least a one-parameter family of embedded examples that are contained in geodesic balls whose radius may be less than, equal to or greater than $π$/2. After explaining the connection to Otsuki tori, we establish lower bounds on the number of immersed free boundary minimal annuli contained in each Otsuki torus in terms of the corresponding rational number. Finally, we show that some of the recent work of Lee and Seo on isoperimetric inequalities and of Lima and Menezes on index bounds extends to geodesic balls equal to or larger than a hemisphere.

Figures (6)

• Figure 1: The sphere $\mathbb{S}^2$, the generating curve $\gamma_a$ (red) for the immersed free boundary minimal annuli $X_a: [-s_i, s_i] \times \mathbb{S}^1 \to \mathbb{S}^3$ ($i = 1, \dots, 4$) and the geodesic spheres $\partial B_{r}^2(p_N)$ (black) that $\gamma_a$ intersects orthogonally, where $a = 0.29$
• Figure 2: Generating curve segments $\gamma_a$ for twenty members of the one-parameter family of embedded free boundary minimal annuli described in Proposition \ref{['prop:one_parameter_family_in_ball']}, where $a$ varies linearly from $-0.49$ (red) to $0.49$ (white)
• Figure 3: Four segments of the generating curve $\gamma_a$ (red) for the Otsuki torus $\Sigma_a(0)$ corresponding to $\frac{p}{q} = \frac{2}{3}$, each generating one of the immersed free boundary minimal annuli described in Theorem \ref{['thm:counting_fbma_in_otsuki_tori']}, together with the geodesic spheres $\partial B_r^2(p_N)$ (black) each segment intersects orthogonally
• Figure 4: Two segments of the generating curve $\gamma_a$ (red) for the Otsuki torus $\Sigma_a(\frac{\pi}{2})$ corresponding to $\frac{p}{q} = \frac{2}{3}$, each generating one of the immersed free boundary minimal annuli described in Theorem \ref{['thm:counting_fbma_in_otsuki_tori']}, together with the geodesic sphere $\mathbb{S}^2 \cap \{x^1 = 0\}$ (black) both segments intersect orthogonally
• Figure 5: Ten segments of the generating curve $\gamma_a$ (red) for the Otsuki torus $\Sigma_a(0)$ corresponding to $\frac{p}{q} = \frac{5}{8}$, each generating one of the immersed free boundary minimal annuli described in Theorem \ref{['thm:counting_fbma_in_otsuki_tori']}, together with the geodesic spheres $\partial B_r^2(p_N)$ (black) each segment intersects orthogonally

Theorems & Definitions (27)

• Proposition 2.1: doCarmo_dajczer_1983
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 3.1