AI 中文总结
研究满足特定条件的初始数据集的彭罗斯猜想，核心方法是利用\(\sigma\) - 逆平均曲率流及单调性公式，在给定条件下证明了边界连通分量的彭罗斯猜想。
AI 中文摘要
设\((M^3, g, \mathbf{k})\)是具有最外层过去表观视界的光滑、连通、渐近平坦的初始数据集。在主导能量条件和\(\mathbf{k}\)的两个最小特征值之和非负的 2 - 凸性条件下，我们证明了边界每个连通分量的彭罗斯猜想。主要工具是\(\sigma\) - 逆平均曲率流和在文献\cite{Dong26SigmaIMCF}中发展的一个单调性公式。
英文摘要
Let $(M^3, g, \mathbf{k})$ be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon $Σ$. We prove the Penrose conjecture, namely that $m_{\mathrm{ADM}}(g) \geq \sqrt{\frac{|Σ|}{16 π}} $, under the assumptions of the dominant energy condition and the $2$-convexity condition that the sum of the two smallest eigenvalues of $\mathbf{k}$ is nonnegative. The main tool is the $σ$-inverse mean curvature flow, together with a monotonicity formula developed in \cite{Dong26SigmaIMCF}.
Commentsv2: connected horizon assumption added; references added;
