Disc stackings and their Morse index

TL;DR

This work constructs free boundary minimal disc stackings in the unit ball with an arbitrary number of layers $N\ge 2$ and large symmetry parameter $m$, yielding varifold convergence to the equatorial disc and precise equivariant Morse index bounds. The approach blends a perturbative gluing method with a two-stage construction: first assemble approximate stackings from $N$ discs connected by $m$ ribbons, then solve a nonlinear PDE via a fixed-point argument, while controlling the cokernel by introducing balancing parameters and dislocations. A detailed spectral analysis of the building blocks (discs and half-catenoidal connectors) via Montiel–Ros theory yields explicit lower and upper bounds on the Morse index and nullity, both absolutely and equivariantly, leading to linear growth in $m$ and new topological realizations. The results imply, in particular, that only for $N=2$ and $N=3$ can stacking surfaces be obtained by one-parameter mountain-pass schemes, whereas realizations for larger $N$ require multi-parameter sweepouts, revealing a sharp variational-perturbative dichotomy with significant implications for free boundary minimal surface theory.

Abstract

We construct free boundary minimal disc stackings, with any number of strata, in the three-dimensional Euclidean unit ball, and prove uniform, linear lower and upper bounds on the Morse index of all such surfaces. Among other things, our work implies for any positive integer $k$ the existence of $k$-tuples of distinct, pairwise non-congruent, embedded free boundary minimal surfaces all having the same topological type. In addition, since we prove that the equivariant Morse index of any such free boundary minimal stacking, with respect to its maximal symmetry group, is bounded from below by (the integer part of) half the number of layers, it follows that any possible realization of such surfaces via an equivariant min-max method would need to employ sweepouts with an arbitrarily large number of parameters. This also shows that it is only for $N=2$ and $N=3$ layers that free boundary minimal disc stackings can be obtained by means of one-dimensional mountain pass schemes.

Figures (10)

• Figure 1: The two cases for the structure of the boundary when attaching a horizontal disc $D$ equivariantly through vertical half-neck ribbons $\mathrm{rib}_k$.
• Figure 2: A free boundary minimal disc stacking with $N=5$ (left image) respectively $N=4$ (right image) and $m=8$. Highlighted is a fundamental domain with respect to the $\mathbb{Z}_m$-action.
• Figure 3: The domains defined in \ref{['eqn:Lambda']}--\ref{['eqn:Lambdatau']} and the image of $\Phi\bigl(\sigma,\theta,\omega^B_{m,h^B}(\sigma,\theta)\bigr)$ defined on $\Lambda^\Phi_m$.
• Figure 4: Profiles of the cutoff functions $\psi^\pm_m,\psi^\sigma_m\colon\Lambda^\Phi_m\to\mathbb{R}$ defined in \ref{['eqn:cat_cutoffs']}.
• Figure 5: The graphs $\Gamma_{m,h^B,\tau_+,h^K_+}$ (left) and $\Gamma_{m,h^B,\tau_+,h^K_+,\tau_-,h^K_-}$ (right). Roughly speaking, the red, green, respectively blue regions are shaped by the first, second respectively third summand of \ref{['angular_height_function_one_bridge']} and \ref{['angular_height_function_two_bridges']}; the yellow region is shaped by the fourth summand of \ref{['angular_height_function_two_bridges']}.

Theorems & Definitions (64)

• Theorem 1.1
• Corollary 1.2: Polymorphism
• proof
• Remark 1.3: Implications for min-max stacking constructions
• Remark 1.4: Index growth in terms of topological data
• Lemma 2.1
• proof
• Lemma 2.2: Geometric estimates on the $K_i$ regions
• proof
• Lemma 2.3: Geometric estimates on the $D_i$ regions