[Paper Review] Positive Scalar Curvature and Minimal Hypersurface Singularities

Richard Schoen, Shing‐Tung Yau|arXiv (Cornell University)|Apr 18, 2017

Geometric Analysis and Curvature FlowsMathematics20 citations

TL;DR

This paper extends the minimal hypersurface method to prove the positive mass theorem and classify manifolds of positive scalar curvature in all dimensions, overcoming singularities via weighted minimal slicings. It establishes that singular sets in such slicings have Hausdorff codimension at least three, enabling the method to work down to dimension 2 and proving the positive mass theorem without spin assumptions.

ABSTRACT

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of \cite{sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.

Motivation & Objective

• To extend the minimal hypersurface method for positive scalar curvature problems beyond dimension 8, where singularities previously obstructed the approach.
• To develop a theory of minimal k-slicings that incorporates singular sets with controlled Hausdorff dimension.
• To prove the positive mass theorem in all dimensions without requiring the spin condition.
• To generalize topological obstructions to positive scalar curvature using cohomological classes and minimal 2-slicings.
• To establish regularity and compactness theorems for weighted minimal slicings in the presence of small singular sets.

Proposed method

• Constructs minimal k-slicings by recursively minimizing weighted volume using positive first eigenfunctions of the second variation form.
• Introduces a modified second variation form $ Q_j $ that is more coercive and remains well-behaved even with small singular sets.
• Uses weighted $ L^2 $ norms and eigenfunctions to define evolving weights $ \rho_j $, ensuring integral control of geometry across slices.
• Applies rescaling and monotonicity theorems (volume and frequency) to analyze singular points and prove that singular sets are homogeneous cones.
• Employs compactness theorems for minimal k-slicings to show that singular sets in $ \Sigma_k $ have Hausdorff dimension at most $ k-3 $.
• Reduces the positive mass theorem to the compact case by capping off ends and using Lohkamp’s observation to construct metrics with nonnegative scalar curvature.

Experimental results

Research questions

• RQ1 Can the minimal hypersurface method be extended to all dimensions despite the presence of singularities in minimal hypersurfaces?
• RQ2 What is the maximal possible Hausdorff dimension of the singular set in a minimal k-slicing, and how does it affect the regularity of lower-dimensional slices?
• RQ3 Can the positive mass theorem be proven in all dimensions without assuming the manifold is spin?
• RQ4 How can cohomological classes in $ H^1(M,\mathbb{Z}) $ be used to obstruct the existence of positive scalar curvature metrics?
• RQ5 What conditions ensure that the final 2-slice in a minimal k-slicing is a union of smooth 2-spheres?

Key findings

• The singular set in any minimal k-slicing has Hausdorff dimension at most $ k-3 $, ensuring regularity in dimensions 1 and 2.
• For any minimal 2-slicing, the 2-dimensional slice $ \Sigma_2 $ is diffeomorphic to a disjoint union of 2-spheres, each satisfying the Gauss-Bonnet theorem.
• The positive mass theorem holds in all dimensions without a spin assumption, with ADM mass nonnegative and zero only if the manifold is isometric to $ \mathbb{R}^n $.
• If $ \alpha_1, \ldots, \alpha_{n-1} \in H^1(M,\mathbb{Z}) $ have nonvanishing intersection with the fundamental class, then $ M $ cannot admit a metric of positive scalar curvature.
• The existence of a minimal 2-slicing with $ \Sigma_2 $ representing a nonzero class in $ H_2(M,\mathbb{Z}) $ implies $ \Sigma_2 $ is a union of smooth 2-spheres, and any other $ \alpha_{n-1} \in H^1(M,\mathbb{Z}) $ must pair trivially with this class.