Variational Effect of Boundary Mean Curvature on ADM Mass in General Relativity.pdf

VARIATIONAL EFFECT OF BOUNDARY MEAN CURVATURE ON ADM MASS IN GENERAL RELATIVITY 3 PENGZI MIAO 0 0 Abstract. We extend the idea and techniques in [14] to study variational 2 effect of the boundary geometry on the ADM mass of an asymptotically flat p manifold. We show that, for a Lipschitz asymptotically flat metric extension of e a bounded Riemannian domain with quasi-convex boundary, if the boundary S mean curvature of the extension is dominated by but not identically equal to the one determined by the given domain, we can decrease its ADM mass while 8 raising its boundary mean curvature. Thus our analysis implies that, for a 1 domain with quasi-convex boundary, the geometric boundary condition holds in Bartnik’s minimal mass extension conjecture [4]. 1 v 5 4 0 1. Introduction 9 0 Asymptotically flat manifolds are often used to model isolated systems in general 3 relativity. A complete Riemannian manifold (Mn , g) with dimension n ≥ 3 is called 0 asymptotically flat if there is a compact set K ⊂ M and a diffeomorphism / h Φ : M \ K → Rn \ {|x| 1} p - such that, in the coordinate chart defined by Φ, h t 2 −p |gij (x) − δij | + |x||gij