Let(M, g)#(M ,g)be a two-sided asymptotically flat hypersurfaceas above. LetΓ֒→Mbe a smooth, compact inner boundary lying on some totallygeodesic hypersurfaceP ֒→Mand assume that, alongΓ,Mis orthogonal toP.Then, if orientations are fixed as above,mg=mh−cn∫Γ〈X, η〉s1(N)dΓ ++cn∫M(2S2ΘX+ Ricg(N,XT))dM,(1.10)wheres1(N)is the mean curvature ofΓ֒→Pwith respect toN,ηis the exteriorunit co-normal toM,S2is the2-mean curvature ofM(see (2.2) below) andXTis the tangential component ofXalongM.

Pstjl(k)=12δi1i2···i2k−3i2k−2stj1j2···j2k−3j2k−2j2k−1j2khj1i1hj2i2···hj2k−2i2k−2gj2k−1jgj2kl,which implies by (2.20) that(4.10)2ePstjl(k)hsj= (2k−1)! (T(2k−1))tpgpl