Free boundary saddle disks in the unit ball

TL;DR

The paper addresses the existence of free boundary saddle disks in the unit ball $\mathbb{B}^3$ that are not equatorial, extending the understanding beyond Nitsche’s equatorial result. It develops a constructive ribbon-based approach: from a planar ribbon $\\gamma$, it builds a free boundary immersion $\\psi_{\\gamma}$ yielding a circle-foliated disk $\\Sigma_{\\gamma}$ in $\\mathbb{B}^3$. By deforming the ribbon to $\\gamma_t$, the authors arrange curvature properties so that the middle piece of the surface becomes saddle, proving the existence of infinitely many $\mathcal{C}^\infty$ free boundary saddle disks not equatorial (typically with self-intersections and non-analytic). The work provides a concrete method to generate saddle disks and raises questions about embeddedness and the analytic case's potential uniqueness. Overall, it advances the geometric understanding of free boundary minimal surfaces and their Weingarten-type generalizations in a canonical domain.

Abstract

We construct an infinite family of non-planar free boundary disks of non-positive Gaussian curvature in the unit ball of $\mathbb{R}^3$.

Figures (5)

• Figure 1.1: Equatorial disk in $\mathbb{B}^3$.
• Figure 1.2: One of the free boundary saddle disks in Theorem \ref{['thm:main']}.
• Figure 2.1: Example of a ribbon $\gamma$.
• Figure 2.2: Piece of a curve $c_p$ inside $\mathbb{B}^3$.
• Figure 3.1: Ribbon $\gamma_t$ for $t = 3/4$.

Theorems & Definitions (18)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Remark 2.8