A note on Killing fields and CMC hypersurfaces

Abstract

In this note we give some sufficient conditions for a CMC-hypersurface in a Riemannian manifold N to be invariant under the 1-parameter group of isometries generated by a Killing field on N . Our main result improves on previous ones by hinges on a new, simple existence theorem for a first zero of solutions of an ODE naturally associated to the problem. This theorem implies some classical oscillation criteria of W. Ambrose and R. Moore. Extension to constant higher-order mean curvature hypersurfaces are also presented.

Stable maximal hypersurfaces in Lorentzian spacetimes

“…By the coarea formula, This, together with (39) implies, by Corollary 2.9 of [29], that z is oscillatory. Let R ≤ R 1 < R 2 be two consecutive zeros of z such that z > 0 on (R 1 , R 2 ).…”

“…Since v k ∈ L ∞ loc (R + 0 ), there exists a solution z of (106) with z ∈ Lip loc (R + 0 ) due to Proposition 4.2 of [11]. Moreover, from the coarea formula and (101) This condition and the fact that v −1 k ∈ L ∞ loc (R + ) and v −1 k / ∈ L 1 (+∞) enable us to use Corollary 2.9 of [29] to obtain that any solution z of (106) is oscillatory. Taking now R ≤ R 1 < R 2 two consecutive zeros of z such that z > 0 on (R 1 , R 2 ) we define the function ϕ(x) := z(r(x)), where r(x) is the distance from x to o in (M, g), and compute…”

“…So, problem (92) has a positive solution and (88) holds. By Theorem 2.8 of [29], for some (hence any) a > 0, where B s is the geodesic ball of (M, g) centered at o with radius s, then either ρ ′′ (τ ) ≡ 0 on M or ρ ′′ (τ ) attains negative values at some points of M .…”