International Congress of Mathematicians

234 Robert BartnikTheorem 2 If (M, g) is asymptotically flat with R(g) > 0 and boundary S ~ S 2with mean curvature satisfyingHn < 2/r (2.4)where r = ^/|£|/47r, then UIADM > 0, with equality iff (M,g) = (R 3 \F(r),5).The proof starts with the Riemannian form of the Schrödinger-Lichnerowicz-Witten identity [48, 35, 53]M(\V'tp\ 2 + \R(g)\'tp\ 2 - \Vip\ 2 )dv M = 4Tx\'tp 00 \ 2 m A DM + f p(ip), (2.5)where ß('ip) is the Nester-Witten form [38]ß(iP) = (^(^ + 1^*5:. (2.6)The boundary condition P + ip\j: = 0 is elliptic and it can be shown [8] there isa spinor on M satisfying Vip = 0 with boundary conditions ip —¥ rpoo ^ 0 asr ^ oo and (2.2) on S. It follows from (2.3) and (2.2) that {tp, (£> s + |F E )i/>) \/|S|/167T, (2.7)A closed minimal surface is said to be an outermost horizon or outer-minimizinghorizon if M contains no least area surfaces homologous to S in the asymptoticregion exterior to S. The outermost condition is essential, since examples of nonnegativescalar curvature manifolds can be constructed by forming the connectedsum of M and large spheres by arbitrarily small and large necks.The Penrose conjecture has been established by Huisken and Ilmanen [29, 30]using a variational level set formulation of the inverse mean curvature flow [23], andby Bray [12] by a very interesting conformai deformation argument. Bray's proofis more general since it takes into account contributions from all the connectedcomponents of the boundary.3. Quasi-local massThus it is natural to consider y/\S\/16w as the mass of a black hole (minimalsurface) S. More generally, the correspondence with Newtonian gravity suggeststhat any bounded region (Q, g) should have a quasi-local mass, which measures bothJT.