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Chern's Conjecture in the Dupin case

Reiko Miyaoka

TL;DR

This work advances Chern's conjecture in the Dupin setting by proving isoparametricity for closed Dupin hypersurfaces under strategic extra conditions: (i) $g=3$ with Dupin and CMC, (ii) $g=4$ with CSC, (iii) $g=4$ with constant Lie curvature, and (iv) $g=6$ with three independent constant Lie curvatures. The authors combine topological tautness with Lie sphere geometry to translate local curvature data into global constancy: for $g=3$ through parallel-hexagon intersections, for $g=4$ via CSC or Lie curvature premises, and for $g=6$ using multi-Lie-curvature constraints to enforce a parallel dodecagon structure along normal geodesics. Central to the method are the notions of curvature distributions, focal data, and Lie transformations that preserve the Lie curvature, enabling reductions to isoparametric models. Collectively, these results narrow the gap toward Chern's conjecture in higher $g$ by showing Dupin plus curvature constancies suffice to force isoparametricity in the studied cases, with several open problems guiding future work. The work also highlights tautness as a global topological tool to bridge local geometric data with global rigidity in the Dupin category.

Abstract

Chern's conjecture states that a closed minimal hypersurface in the euclidean sphere is isoparametric if it has constant scalar curvature. When the number $g$ of distinct principal curvatures is greater than three, few satisfactory results have been known. We attack the conjecture in the Dupin hypersurface case. Our results are: A closed proper Dupin hypersurface with constant mean curvature is isoparametric (i) if $g=3$, (ii) if $g=4$ and has constant scalar curvature, or (iii) if $g=4$ and has constant Lie curvature, and (iv) if $g=6$ and has constant Lie curvatures. These cover all the non-trivial cases for a closed proper Dupin to be isoparametric since $g$ can take only values $1,2,3,4,6$. The originality of the proof is a use of topology and geometry, which reduces assumptions needed in the algebraic argument.

Chern's Conjecture in the Dupin case

TL;DR

This work advances Chern's conjecture in the Dupin setting by proving isoparametricity for closed Dupin hypersurfaces under strategic extra conditions: (i) with Dupin and CMC, (ii) with CSC, (iii) with constant Lie curvature, and (iv) with three independent constant Lie curvatures. The authors combine topological tautness with Lie sphere geometry to translate local curvature data into global constancy: for through parallel-hexagon intersections, for via CSC or Lie curvature premises, and for using multi-Lie-curvature constraints to enforce a parallel dodecagon structure along normal geodesics. Central to the method are the notions of curvature distributions, focal data, and Lie transformations that preserve the Lie curvature, enabling reductions to isoparametric models. Collectively, these results narrow the gap toward Chern's conjecture in higher by showing Dupin plus curvature constancies suffice to force isoparametricity in the studied cases, with several open problems guiding future work. The work also highlights tautness as a global topological tool to bridge local geometric data with global rigidity in the Dupin category.

Abstract

Chern's conjecture states that a closed minimal hypersurface in the euclidean sphere is isoparametric if it has constant scalar curvature. When the number of distinct principal curvatures is greater than three, few satisfactory results have been known. We attack the conjecture in the Dupin hypersurface case. Our results are: A closed proper Dupin hypersurface with constant mean curvature is isoparametric (i) if , (ii) if and has constant scalar curvature, or (iii) if and has constant Lie curvature, and (iv) if and has constant Lie curvatures. These cover all the non-trivial cases for a closed proper Dupin to be isoparametric since can take only values . The originality of the proof is a use of topology and geometry, which reduces assumptions needed in the algebraic argument.

Paper Structure

This paper contains 14 sections, 32 theorems, 189 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Closed case: $g=3$
  4. Closed CMC and CSC Dupin with $g=4$
  5. Review of the Lie sphere geometry
  6. Closed CMC and CLC Dupin with $g=4$ (iii)
  7. Critical point of all the principal curvatures
  8. Conformal transformation
  9. Closed case: $g=6$ (iv)
  10. Strategy for $g=6$
  11. Critical point of principal curvatures
  12. Angle relation and the Lie curvature
  13. Conformal transformation
  14. Problems

Key Result

Theorem 1.1

A closed CMC Dupin hypersurface $M$ in $S^{n}$ is isoparametric

Figures (11)

  • Figure 1: Link of leaves
  • Figure 2: Angles
  • Figure 3: Parallel hexagon
  • Figure 4: Link of leaves
  • Figure 5: Angles
  • ...and 6 more figures

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Lemma 4.6
  • ...and 22 more