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On the index of minimal 2-tori in the 4-sphere

Rob Kusner, Peng Wang

Abstract

In this note we prove that any minimal $2$-torus in $S^4$ has Morse index at least $6$, with equality if and only if it is congruent to the Clifford torus in some great $S^3\subset S^4$.For a minimal $2$-torus in $S^n$ with vanishing Hopf differential, we show that its index is at least $n+3$, and that this estimate is sharp: the equilateral $2$-torus fully embedded in $S^5\subset S^n$ as a homogeneous minimal surface in $S^n$ has index exactly $n+3$.

On the index of minimal 2-tori in the 4-sphere

Abstract

In this note we prove that any minimal -torus in has Morse index at least , with equality if and only if it is congruent to the Clifford torus in some great .For a minimal -torus in with vanishing Hopf differential, we show that its index is at least , and that this estimate is sharp: the equilateral -torus fully embedded in as a homogeneous minimal surface in has index exactly .

Paper Structure

This paper contains 17 sections, 14 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Acknowledgements
  4. Minimal surfaces in $S^n$
  5. Variation formulas for the Area functional and the Morse index
  6. Computations in a complex coordinate and Calabi-Hopf differentials
  7. A Simons-type identity for the Calabi differential
  8. Further remarks concerning the Jacobi eigenfields of Simons
  9. Minimal tori in $S^n$
  10. Technical details on the Calabi-Hopf differentials
  11. The case with nonvanishing Hopf differential $\mathcal{H}$
  12. The index estimate
  13. The superminimal case of vanishing Hopf differential $\mathcal{H}$
  14. Remarks on the higher genus case and the nonorientable case
  15. The case with vanishing Hopf differential $\mathcal{H}$
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

Let $f:T^2\rightarrow S^4$ be a minimal torus in the unit $4$-sphere. Then we have $\operatorname{Ind}(f)\geq 6$ and equality holds if and only if $f$ is congruent to the Clifford torus in some great $S^3\subset S^4$.

Theorems & Definitions (25)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • ...and 15 more