## Proof of the Riemannian Penrose inequality using the positive mass theorem.(English)Zbl 1039.53034

An asymptotically flat 3-manifold is a Riemannian manifold $$(M^3, g)$$ which, outside a compact set, is a disjoint union of one ore more regions (called ends) diffeomorphic to $$({\mathbb R}^3\setminus B_1(0), \delta)$$, where the metric $$g$$ in each of $${\mathbb R}^3$$ coordinate charts approaches the standart metric $$\delta$$ on $${\mathbb R}^3$$ at infinity. The positive mass theorem and the Penrose conjecture are both statements which refer to a particular chosen end of $$(M^3, g)$$. The total mass of $$(M^3, g)$$ is a parameter related to how fast this chosen end of $$(M^3, g)$$ becomes flat at infinity. The main result of the paper is the proof of the following geometric statement – the Riemannian Penrose conjecture: Let $$(M^3, g)$$ be a complete, smooth, asymptotically flat 3-manifold with nonnegative scalar curvature and total mass $$m$$ whose outermost minimal spheres have total surface area $$A$$. Then $$m\geq\sqrt{\frac{A}{16\pi}}$$ with equality if and only if $$(M^3, g)$$ is isometric to the Schwarzschild metric $$({\mathbb R}^3\setminus\{0\}, s)$$ of mass $$m$$ outside their respective horizons.

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53Z05 Applications of differential geometry to physics
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