Positivity of mass for certain spacetimes with horizons

Assuming the existence of a solution of the 3-harmonic equation Del i( mod Del rho mod 3 Del i rho )=0 on a spacelike 3-manifold Sigma with boundary K(assume that the solution has no critical points and fulfils certain asymptotic conditions), the author constructs an integral identity. The boundary term (at infinity) of this integrand is ADM mass at infinity. The global integrand, in the case when Sigma is a maximal surface with spacetime curvature satisfying the weak energy condition, is manifestly positive. The boundary term on K gives some number (when integrated over K) related to the area of the horizon but in general differs from it. This gives a proof of a following theorem (which is similar to the Penrose inequality for a time symmetric surface Sigma ).