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Minimal surfaces and comparison geometry

Otis Chodosh

TL;DR

This survey demonstrates how area-minimizing hypersurfaces and their variational descendants yield powerful comparison-geometric conclusions about scalar curvature and related curvature conditions. It develops and canvasses a suite of techniques—first/second variation, spectral positivity, $\mu$-bubble localization, and generic regularity—that together constrain topology and geometry under positive scalar curvature, including aspherical (K(π,1)) manifolds, noncompact settings, and Bernstein-type problems. Key contributions include dimensionally robust inductive-descent arguments, localized curvature control via $\mu$-bubbles, and recent advances in generic regularity that mitigate singularities in high dimensions. The results have broad implications for understanding when PSC metrics can exist, the structure of manifolds under curvature constraints, and the interplay between geometric analysis, topology, and global Riemannian geometry. Collectively, the work advances both classical conjectures and modern extensions across dimensions, providing a cohesive framework that links variational minimal-surface techniques with topological classifications and curvature-positivity phenomena.

Abstract

We discuss applications of minimal surfaces to comparison geometry.

Minimal surfaces and comparison geometry

TL;DR

This survey demonstrates how area-minimizing hypersurfaces and their variational descendants yield powerful comparison-geometric conclusions about scalar curvature and related curvature conditions. It develops and canvasses a suite of techniques—first/second variation, spectral positivity, -bubble localization, and generic regularity—that together constrain topology and geometry under positive scalar curvature, including aspherical (K(π,1)) manifolds, noncompact settings, and Bernstein-type problems. Key contributions include dimensionally robust inductive-descent arguments, localized curvature control via -bubbles, and recent advances in generic regularity that mitigate singularities in high dimensions. The results have broad implications for understanding when PSC metrics can exist, the structure of manifolds under curvature constraints, and the interplay between geometric analysis, topology, and global Riemannian geometry. Collectively, the work advances both classical conjectures and modern extensions across dimensions, providing a cohesive framework that links variational minimal-surface techniques with topological classifications and curvature-positivity phenomena.

Abstract

We discuss applications of minimal surfaces to comparison geometry.

Paper Structure

This paper contains 15 sections, 33 theorems, 29 equations.

Table of Contents

  1. Introduction.
  2. Minimization of area.
  3. First and second variation.
  4. The Geroch conjecture
  5. Inductive descent.
  6. Spectral positivity of curvature.
  7. Localization via prescribed mean curvature.
  8. The $K(\pi,1)$ conjecture.
  9. The geometry of positive scalar curvature.
  10. Non-compact manifolds with positive scalar curvature.
  11. Non-strictly positive scalar curvature
  12. Other results.
  13. Other curvature conditions
  14. The (stable) Bernstein problem.
  15. Generic regularity for the Plateau problem.

Key Result

Theorem 1

Let $(M^{n+1},g)$ be a closed Riemannian manifold and $\sigma \in H_n(M;\mathbb{Z})$ be a codimension-one homology class. If $n+1\leq 7$ then there's a smooth representative $\Sigma \in \sigma$ of least area.

Theorems & Definitions (55)

  • Theorem 1: cf. Simon:GMT
  • Theorem 2: SY:3d-torus
  • proof
  • Theorem 3: SY:3d-torus
  • proof
  • Theorem 4: SY:manuscriptaGromovLawson
  • proof : Sketch of the proof
  • Remark 5
  • Theorem 6: Gromov:metric-inequalitiesgromov1996positivegromov2019lectureszhu2020width
  • proof : Sketch of the proof for $n=2$
  • ...and 45 more