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On the topology of stable minimal hypersurfaces in a homeomorphic $S^4$

Chao Li, Boyu Zhang

TL;DR

This work develops a framework to control the topology of stable minimal hypersurfaces in PSC 4-manifolds with boundary that embed in a space homeomorphic to $S^4$. By combining a White-inspired extremal-position minimization with covering-space arguments and 4-manifold topology, the authors show that minimal hypersurfaces homologous to a boundary component have constrained topology, ruling out nontrivial spherical space-forms in their prime decomposition. They derive a black-hole topology theorem: the outermost horizons in suitable asymptotically flat 4-manifolds are diffeomorphic to unions of $S^3$ or connected sums of $S^2\times S^1$, with stronger results under positive bi-Ricci curvature. The paper also extends these ideas to Plateau problems in homeomorphic $D^4$, constructs PSC embeddings involving lens spaces, and discusses the Dahl–Larsson horizon examples, highlighting both geometric-analytic methods and topological obstructions with potential applications to general relativity.

Abstract

We construct stable minimal hypersurfaces with simple topology in certain compact $4$-manifolds $X$ with boundary, where $X$ embeds into a smooth manifold homeomorphic to $S^4$. For example, if $X$ is equipped with a Riemannian metric $g$ with positive scalar curvature, we prove the existence of a stable minimal hypersurface $M$ that is diffeomorphic to either $S^3$ or a connected sum of $S^2\times S^1$'s, ruling out spherical space forms in its prime decomposition. These results imply new theorems on the topology of black holes in four dimensions. The proof involves techniques from geometric measure theory and $4$-manifold topology.

On the topology of stable minimal hypersurfaces in a homeomorphic $S^4$

TL;DR

This work develops a framework to control the topology of stable minimal hypersurfaces in PSC 4-manifolds with boundary that embed in a space homeomorphic to . By combining a White-inspired extremal-position minimization with covering-space arguments and 4-manifold topology, the authors show that minimal hypersurfaces homologous to a boundary component have constrained topology, ruling out nontrivial spherical space-forms in their prime decomposition. They derive a black-hole topology theorem: the outermost horizons in suitable asymptotically flat 4-manifolds are diffeomorphic to unions of or connected sums of , with stronger results under positive bi-Ricci curvature. The paper also extends these ideas to Plateau problems in homeomorphic , constructs PSC embeddings involving lens spaces, and discusses the Dahl–Larsson horizon examples, highlighting both geometric-analytic methods and topological obstructions with potential applications to general relativity.

Abstract

We construct stable minimal hypersurfaces with simple topology in certain compact -manifolds with boundary, where embeds into a smooth manifold homeomorphic to . For example, if is equipped with a Riemannian metric with positive scalar curvature, we prove the existence of a stable minimal hypersurface that is diffeomorphic to either or a connected sum of 's, ruling out spherical space forms in its prime decomposition. These results imply new theorems on the topology of black holes in four dimensions. The proof involves techniques from geometric measure theory and -manifold topology.

Paper Structure

This paper contains 11 sections, 16 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Strategy of proof
  3. A theorem on the topology of black holes
  4. Organization of the paper
  5. Stable minimal hypersurfaces of extremal positions
  6. Eliminating space forms
  7. A black hole topology theorem
  8. Extensions, examples and further questions
  9. Topology of the solution to Plateau problems in $D^4$
  10. A PSC embedding of connected sums of lens spaces
  11. The Dahl-Larsson example

Key Result

Theorem 1.2

Suppose $(X^4,g)$ is a smooth Riemannian manifold satisfying topo.assumption.on.domain, such that $g$ has positive scalar curvature and $\partial X$ is $g$-weakly mean convex. Then $(X^4,g)$ contains a two-sided embedded stable minimal hypersurface that is diffeomorphic to $S^3$ or a connected sum o

Theorems & Definitions (40)

  • Remark 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 30 more