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Bifurcations of Clifford tori in ellipsoids

Renato G. Bettiol, Paolo Piccione

Abstract

We prove that 3-dimensional ellipsoids invariant under a 2-torus action contain infinitely many distinct immersed minimal tori, with at most one exception. These minimal tori bifurcate from the 2-torus orbit of largest volume at a dense set of eccentricities, and remain invariant under a circle.

Bifurcations of Clifford tori in ellipsoids

Abstract

We prove that 3-dimensional ellipsoids invariant under a 2-torus action contain infinitely many distinct immersed minimal tori, with at most one exception. These minimal tori bifurcate from the 2-torus orbit of largest volume at a dense set of eccentricities, and remain invariant under a circle.
Paper Structure (12 sections, 8 theorems, 24 equations, 2 figures)

This paper contains 12 sections, 8 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Geometric setup
  4. Symmetry reduction
  5. Geodesics in the renormalized orbit space
  6. Geodesic equation
  7. Clifford geodesic
  8. Orthogonal geodesics
  9. Bifurcations of the Clifford geodesic
  10. Local bifurcation
  11. Bifurcation branches
  12. Global behavior

Key Result

Proposition 2.1

An $\{1\}\times\mathsf{S}^1$-invariant surface $\Sigma$ in $\mathds{S}^3_a$ is minimal if and only if its quotient $\Sigma/\mathsf{S}^1$ is a geodesic in $\Omega_{a}$. In particular, $\Sigma$ is an $\{1\}\times\mathsf{S}^1$-invariant minimal torus in $\mathds{S}^3_a$ if and only if $\Sigma/\mathsf{S

Figures (2)

  • Figure 1: Schematic depiction of $\Omega_a$ with Clifford geodesic $\gamma_{a,0}$ in red, and portion of a geodesic $\gamma_{a,s}$, $s<0$, in orange.
  • Figure 2: Schematic illustration depicting qualitative behavior of bifurcation branches $\mathcal{B}_{(j,k)}$ for small $k$, including $\mathcal{B}_{(1,1)}$ in blue, according to whether $\lim_{t\to+\infty} a^j_k(t)=0$ for all $j,k$ (top) or not (bottom). The expected (conjectural) behavior is the top illustration. These images are not the result of any numerical experiment.

Theorems & Definitions (27)

  • Proposition 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Remark 3.6
  • ...and 17 more