Splitting of 3-manifolds and rigidity of area-minimising surfaces

Abstract

In this paper we modify an argument of Bray, Brendle and Neves to prove an area comparison result (Theorem 2) for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature.This theorem is a variant of a comparison theorem (Theorem 3.2 (d) in the 1978 paper) of Heintze and Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker, but the assumptions on the surface are considerably more restrictive.We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature that were first proved

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

“…It is then straightforward to check that the normal vector fields of Σ ρ is parallel (see [BBN10] or [MM15]). In particular, its flow is a flow by isometries and therefore provides the local splitting.…”

Rigidity of marginally outer trapped (hyper)surfaces with negative 𝜎-constant

“…Before passing to the main result of this section, we are going to prove a calculus lemma and to present the definition of n-convexity. The proof of the following lemma is based on the techniques presented in [19]. ([0, ε)) and η, ξ, ρ ∈ C 0 ([0, ε)) be functions such that max{f, ρ} ≥ 0, ξ ≥ 0, η > 0, f (0) = 0, and…”

“…Partially motivated by some issues concerning the topology of back holes, Cai and Galloway [8] solved the problem posed by Fischer-Colbrie and Schoen, in a paper which has inspired a great deal of subsequent works in the subject (e.g. [1,6,19,21]). They proved: Theorem 1.2 (Cai-Galloway).…”

Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds

“…The same technique has been used by many authors in the literature (e.g. [1,4,6,11]). for all x ∈ Σ and t ∈ (−ε, ε).…”