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Nonplanar minimal spheres in ellipsoids of revolution

Renato G. Bettiol, Paolo Piccione

TL;DR

The paper addresses the Yau minimal-sphere problem in elongated ellipsoids by leveraging a symmetry-enabled reduction to geodesics in an orbit space and applying global bifurcation theory. It proves that, under a symmetry b=c, there are arbitrarily many nonplanar embedded minimal 2-spheres in E(a,b,b,d) for large elongation, with a linear growth lower bound on their count and detailed asymptotics: each new surface S_m(a) converges to the limit cylinder Σ_1(∞) with multiplicity m and increasing Morse index. The authors develop a singular Sturm–Liouville spectral framework to locate bifurcation instants a_m, use Crandall–Rabinowitz and Rabinowitz theorems to generate global bifurcation branches, and show that the resulting surfaces exhibit a scarring-like behavior on the limiting stable minimal sphere. This yields both quantitative growth results and geometric descriptions of how nonplanar minimal spheres proliferate and concentrate near the cylinder as elongation grows, with implications for minimal-surface theory in varying ambient geometries.

Abstract

We use global bifurcation techniques to establish the existence of arbitrarily many geometrically distinct nonplanar embedded smooth minimal 2-spheres in sufficiently elongated 3-dimensional ellipsoids of revolution. More precisely, we quantify the growth rate of the number of such minimal spheres, and describe their asymptotic behavior as the ellipsoids converge to a cylinder.

Nonplanar minimal spheres in ellipsoids of revolution

TL;DR

The paper addresses the Yau minimal-sphere problem in elongated ellipsoids by leveraging a symmetry-enabled reduction to geodesics in an orbit space and applying global bifurcation theory. It proves that, under a symmetry b=c, there are arbitrarily many nonplanar embedded minimal 2-spheres in E(a,b,b,d) for large elongation, with a linear growth lower bound on their count and detailed asymptotics: each new surface S_m(a) converges to the limit cylinder Σ_1(∞) with multiplicity m and increasing Morse index. The authors develop a singular Sturm–Liouville spectral framework to locate bifurcation instants a_m, use Crandall–Rabinowitz and Rabinowitz theorems to generate global bifurcation branches, and show that the resulting surfaces exhibit a scarring-like behavior on the limiting stable minimal sphere. This yields both quantitative growth results and geometric descriptions of how nonplanar minimal spheres proliferate and concentrate near the cylinder as elongation grows, with implications for minimal-surface theory in varying ambient geometries.

Abstract

We use global bifurcation techniques to establish the existence of arbitrarily many geometrically distinct nonplanar embedded smooth minimal 2-spheres in sufficiently elongated 3-dimensional ellipsoids of revolution. More precisely, we quantify the growth rate of the number of such minimal spheres, and describe their asymptotic behavior as the ellipsoids converge to a cylinder.
Paper Structure (28 sections, 33 theorems, 92 equations, 4 figures)

This paper contains 28 sections, 33 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Overview of proofs
  4. Acknowledgements
  5. Preliminaries
  6. Hsiang--Lawson reduction
  7. Bifurcation theory
  8. Local bifurcation
  9. Global bifurcation
  10. Disjoint branches
  11. Minimal spheres of revolution
  12. Symmetry reduction
  13. Orthogonal geodesics
  14. Even and odd geodesics
  15. Jacobi equation of the trivial (planar) solution
  16. ...and 13 more sections

Key Result

Theorem 2.1

A $\mathsf{G}$-invariant submanifold $\Sigma$ of cohomogeneity $k\geq1$ is minimal in $(M,\mathrm g)$ if and only if the projection $\Sigma_{{\rm{pr}}}/\mathsf{G}=\Pi(\Sigma\cap M_{\rm{pr}})$ of its principal part is minimal in $(M_{{\rm{pr}}}/\mathsf{G},V^{2/k}\check{\mathrm g})$.

Figures (4)

  • Figure 1: Schematic depiction of $\Omega_a$, with boundary in red, parametrized by \ref{['eq:beta']}, and free boundary geodesics $\gamma_{\rm{ver}}$ and $\gamma_{\rm{hor}}$, with endpoints $\beta\left(\frac{\pi}{2}\right)$ and $\beta\left(-\frac{\pi}{2}\right)$, respectively $\beta(0)$ and $\beta(\pi)$.
  • Figure 2: Schematic representation of an even geodesic (blue), and an odd geodesic (green) in $\Omega_a$.
  • Figure 3: Schematic representation of eigenvalues of $(\mathcal{P}_a)_{\rm{even}}$ and $(\mathcal{P}_a)_{\rm{odd}}$ as $a>0$ varies, for fixed $b=c>0$ and $d>0$.
  • Figure 4: Schematic illustration of trivial branch $\mathcal{B}_{\rm{triv}}$ in red; and bifurcation branches $\mathfrak B_m$, $m\geq1$, in the upper picture. In the lower picture, these branches are individuated as $\mathfrak B_n^{\rm{even}}$ and $\mathfrak B_n^{\rm{odd}}$.

Theorems & Definitions (82)

  • Theorem 2.1: Hsiang--Lawson
  • Theorem 2.2: Crandall--Rabinowitz
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Rabinowitz
  • Remark 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Proposition 3.1
  • ...and 72 more