We prove that the Riemannian Penrose inequality holds for asymptotically flat 3‐manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the mass being a well‐defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential‐theoretic version of it, recently introduced by Agostiniani, Oronzio, and the third named author. As a consequence, we establish the equality between mass and Huisken's isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose inequality in terms of the isoperimetric mass on any 3‐manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well‐posed notion of weak inverse mean curvature flow (IMCF). In particular, such isoperimetric Riemannian Penrose inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's isoperimetric mass and the Hawking mass.

中文翻译：

---

关于等周长黎曼彭罗斯不等式


我们证明，黎曼彭罗斯不等式适用于具有非负标量曲率和相连水平边界的渐近平坦的 3 流形，前提是满足最佳衰减假设，这导致质量是定义明确的几何不变量。我们的证明建立在霍金质量与其潜在理论版本之间的新颖相互作用之上，最近由 Agostiniani、Oronzio 和第三位命名作者提出。因此，我们在上述尖锐假设下建立了质量与 Huisken 等周长质量之间的相等性。此外，我们根据任何具有非负标量曲率、相连水平边界的 3 流形上的等周长质量建立了黎曼彭罗斯不等式，并支持弱反平均曲率流 （IMCF） 的适定概念。特别是，这种等长黎曼彭罗斯不等式不需要流形的渐近平坦度。该论点基于一个新的渐近比较结果，该结果涉及 Huisken 的等周长质量和 Hawking 质量。