Rigidity and compactness with constant mean curvature in warped product manifolds

A Assumption (H3)* and models in general relativity

In this appendix, we check that the Reissner–Nordstrom manifolds satisfy assumption (H3)*, while the de Sitter–Schwarzschild manifolds do not. This simple fact, combined with the analysis of equality cases in Brendle’s Heintze–Karcher-type inequalities, shows that a stronger stability mechanism for almost-CMC hypersurfaces is at a play in the R–N manifolds. Indeed, when (H3)* holds, Brendle’s argument also provides, in addition to almost-umbilicality, a direct control on the oscillation of the normals with respect to the radial directions as measured by g N ⁢ ( ν Ω , ν Ω ) ; see, for example, condition (4.4).

Let us set G = S n − 1 × ( s 1 , s 2 ) for some 0 < s 1 < s 2 ≤ + ∞ . The dS–S manifold is then ( G , g dSS ) , where

with

When κ > 0 , the upper bound on 𝜅 guarantees that 1 − m ⁢ s 2 − n − κ ⁢ s 2 has exactly two zeros s 1 < s 2 on ( 0 , ∞ ) , while if κ ≤ 0 , we set s 2 = + ∞ and s 1 is the unique zero of 1 − m ⁢ s 2 − n − κ ⁢ s 2 on ( 0 , ∞ ) . The R–N manifold is defined instead as ( G , g RN ) , where

In this case, s 1 is the largest of the two solutions of 1 − m ⁢ s 2 − n + q 2 ⁢ s 4 − 2 ⁢ n = 0 on ( 0 , ∞ ) , while we set s 2 = + ∞ . Both examples can be modeled as g ω = ( 1 / ω ⁢ ( s ) ) ⁢ d ⁢ s ⊗ d ⁢ s + s 2 ⁢ g S n − 1 for a smooth function ω : ( s 1 , s 2 ) → ( 0 , ∞ ) . We then define

so that h ⁢ ( F ⁢ ( s ) ) = s for every s ∈ ( s 1 , s 2 ) , and

Setting M = S n − 1 × ( 0 , r ̄ ) , the map ϕ : M → G defined by ϕ ⁢ ( τ , t ) = ( τ , h ⁢ ( t ) ) is such that

so that ( G , g ω ) is isometric to ( M , g ) , and rigidity of CMC-hypersurfaces in dS–S and R–N manifolds can be studied in their ( M , g ) representations. Since (H0) holds with ρ = 1 , we have

for every s ∈ ( s 1 , s 2 ) . We thus find, in the case of g dSS , where ω ⁢ ( s ) = 1 − m ⁢ s 2 − n − κ ⁢ s 2 ,

so that (H3) holds (but not (H3)*); in the case of g RN , where ω ⁢ ( s ) = 1 − m ⁢ s 2 − n + q 2 ⁢ s 4 − 2 ⁢ n , we have an identical cancellation of the mass term, but thanks to the q 2 -term, we rather find

so that ℓ ⁢ ( s ) is strictly increasing on ( s 1 , s 2 ) , and (H3)* holds. ∎