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Complete stable minimal hypersurfaces in positively curved 4-manifolds

Otis Chodosh, Chao Li, Douglas Stryker

Abstract

We show that the combination of non-negative sectional curvature (or $2$-intermediate Ricci curvature) and strict positivity of scalar curvature forces rigidity of complete (non-compact) two-sided stable minimal hypersurfaces in a $4$-manifold with bounded curvature. In particular, this implies the nonexistence of complete two-sided stable minimal hypersurface in a closed $4$-manifold with positive sectional curvature. Our work leads to new comparison results. We also construct various examples showing rigidity of stable minimal hypersurfaces can fail under other curvature conditions.

Complete stable minimal hypersurfaces in positively curved 4-manifolds

Abstract

We show that the combination of non-negative sectional curvature (or -intermediate Ricci curvature) and strict positivity of scalar curvature forces rigidity of complete (non-compact) two-sided stable minimal hypersurfaces in a -manifold with bounded curvature. In particular, this implies the nonexistence of complete two-sided stable minimal hypersurface in a closed -manifold with positive sectional curvature. Our work leads to new comparison results. We also construct various examples showing rigidity of stable minimal hypersurfaces can fail under other curvature conditions.
Paper Structure (30 sections, 20 theorems, 81 equations)

This paper contains 30 sections, 20 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Concerning the assumption of non-negative sectional curvature
  4. Topological applications
  5. Idea of the proof and related results
  6. Acknowledgements
  7. Organization of the paper
  8. Curvature Preliminaries
  9. The $\mathop{\mathrm{Ric}}\nolimits_{2}\geq0$ Condition
  10. Weakly Bounded Geometry
  11. Preliminary Results on Ends
  12. Ends adapted to geodesic balls
  13. Parabolicity and nonparabolicity
  14. Nonparabolic Ends of Stable Hypersurfaces in 4-Manifolds with $\mathop{\mathrm{Ric}}\nolimits_{2}\geq 0$
  15. Schoen-Yau inequality
  16. ...and 15 more sections

Key Result

Theorem 1.3

If $(X^{4},g)$ is a complete $4$-manifold with weakly bounded geometry, non-negative sectional curvature, and strictly positive scalar curvature $R_g\geq R_{0}>0$, then any complete two-sided stable minimal immersion $M^{3}\to(X^{4},g)$ is totally geodesic and has $\mathop{\mathrm{Ric}}\nolimits_{g}

Theorems & Definitions (55)

  • Example 1.1: Non-compact stable minimal hypersurface in $4$-manifold with $R_{g}\geq 1$
  • Example 1.2: Stable minimal hypersurface in a $4$-manifold with positive sectional curvature
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Example 1.9: Stable minimal hypersurface in a $4$-manifold with strictly positive Ricci curvature
  • Theorem 1.10
  • ...and 45 more