Applications to Positive Scalar Curvature and General Relativity

We now turn to applications where stable minimal hypersurfaces are used as probes of scalar curvature. The guiding mechanism is the Schoen–Yau dimension-reduction argument [SY79a, SY79b]: a stable minimal hypersurface inside a manifold with a scalar-curvature lower bound inherits, after a conformal change, a related lower bound in one lower dimension. This simple idea has two closely connected roles. In positive scalar curvature it leads to topological obstructions such as Geroch’s conjecture. In mathematical general relativity it becomes a tool for proving mass inequalities, beginning with the positive mass theorem and, in the negatively curved setting, the Horowitz–Myers conjecture.

We first record the conformal calculation behind the descent argument, then explain how it proves the positive-scalar-curvature obstruction for tori. The last two sections reinterpret the same method in general relativity: the asymptotically flat case leads to the positive mass theorem, while the asymptotically locally hyperbolic toroidal case leads to the Horowitz–Myers mass inequality. A convenient modern reference for the positive-scalar-curvature part is Chodosh’s notes Stable minimal surfaces and positive scalar curvature.

Sections

• The conformal Laplacian
• Conformal descent on a stable minimal hypersurface
• Geroch’s Conjecture by Dimension Reduction
• The Positive Mass Theorem and the Reduction to Geroch’s Conjecture
• The Horowitz–Myers Conjecture