Two-ended $r$-minimal hypersurfaces in Euclidean space

Extrinsic Black Hole Uniqueness in Pure Lovelock Gravity

“…The key technical ingredient in the proof of Theorem 1 is a remarkable result by (Theorem 3.1 Araújo and Leite ( 2012)), which extends previous contributions in de Lima and Sousa (2011), Hounie and Leite (1999), Schoen (1983). More precisely, they classify complete, embedded two-ended hypersurfaces M → R n+1 such that: a. its 2k-mean curvature vanishes (σ 2k (A) = 0); b. it is elliptic in the sense that σ 2k+1 (A) = 0 everywhere; c. each of its ends is asymptotically rotationally symmetric (they can be be written as a graph over the complement of a ball in a hyperplane Π → R n+1 associated to a function, say u, which behaves precisely as in ( 20)-( 24), depending on the value of q = q k,n ).…”