International Congress of Mathematicians

Black Holes and the Penrose Inequality in General Relativity 263for all t > 0. Then since (M 3 ,g t ) converges to a Schwarzschild metric (in anappropriate sense) which gives equality in the Riemannian Penrose Inequality asdescribed in the introduction,which proves the Riemannian Penrose Inequality for the original metric (M 3 ,g 0 ).The hard part, then, is to find a flow of metrics which preserves nonnegative scalarcurvature and the area of the horizon, decreases total mass, and converges to aSchwarzschild metric as t goes to infinity.2.1. The definition of the flowIn fact, the metrics gt will all be conformai to go- This conformai flow ofmetrics can be thought of as the solution to a first order o.d.e. in t defined byequations11, 12, 13, and 14. Letand UQ(X) = 1. Given the metric gt, defineg t = utixfgo (11)Ti(t) = the outermost minimal area enclosure of S 0 in (M 3 ,g t ) (12)where So is the original outer minimizing horizon in (M 3 ,go). In the cases in whichwe are interested, £(£) will not touch S 0 , from which it follows that £(£) is actuallya strictly outer minimizing horizon of (M 3 ,g t ). Then given the horizon £(£), definevt(x) such thatA go vt(x) = 0 outside £(£)vt(x) = 0 on £(f) (13)lim^oot^x) = -e - *and vt(x) = 0 inside £(£). Finally, given vt(x), defineut(x) = 1 + / v s (x)ds (14)Joso that ut(x) is continuous in t and has uo(x) = 1-Note that equation 14 implies that the first order rate of change of ut(x) isgiven by Vt(x). Hence, the first order rate of change of gt is a function of itself, ga,and vt(x) which is a function of go, t, and £(£) which is in turn a function of gt andS 0 . Thus, the first order rate of change of gt is a function of t, gt, go, and S 0 .Theorem 2 Taken together, equations 11, 12, 13, and 14 define a first ordero.d.e. in t for ut(x) which has a solution which is Lipschitz in the t variable, C 1 inthe x variable everywhere, and smooth in the x variable outside S(£). Furthermore,£(£) is a smooth, strictly outer minimizing horizon in (M 3 ,g t ) for all t > 0, and£(i 2 ) encloses but does not touch S(£i) for all t2 > h > 0.