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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mn>4</m:mn> </m:msup> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> S^{3} or a connected sum of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>S</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msup> <m:mi>S</m:mi> <m:mn>1</m:mn> </m:msup> </m:mrow> </m:math> 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.