AI 中文总结
该研究针对满足特定条件的真嵌入平均凸平面曲面，证明了无视界限制的外在彭罗斯不等式，明确了等号成立的情形及对应几何结构。
AI 中文摘要
设$S\subset\mathbb R^3$是一个具有有限个端点的真嵌入平均凸平面曲面，指定其中一个端点为渐近平坦的，假设$H_S$在该端点可积，并记其外在质量为$m_+(S)$。令$A_S$为将指定端点与所有其他端点分隔开的紧致曲面面积的下确界。我们证明$m_+(S)\geq\sqrt{\frac{A_S}{\pi}}$。该证明未假设存在最外层自由边界极小曲面，也未对其他端点施加渐近或可积性条件。若$A_S=0$，等号仅在欧几里得半空间中成立；完整悬链面在$A_S>0$时实现等号；反之，若$A_S>0$时等号成立，则$A_S$由朝向指定端点的最外层平坦自由边界圆盘达到，且$S$的对应分支为半悬链面。
英文摘要
Let $S\subset\mathbb R^3$ be a properly embedded mean-convex planar surface with finitely many ends. Designate one end as asymptotically flat, assume that $H_S$ is integrable there, and denote its extrinsic mass by $m_+(S)$. Let $A_S$ be the infimum of the areas of compact surfaces separating the distinguished end from all the others. We prove \[ m_+(S)\geq\sqrt{\frac{A_S}π}. \] No outermost free-boundary minimal surface is assumed, and no asymptotic or integrability condition is imposed on the other ends. If $A_S=0$, equality holds precisely for the Euclidean half-space. The complete catenoid realizes equality with $A_S>0$. Conversely, if equality holds with $A_S>0$, then $A_S$ is attained by a flat free-boundary disk that is outermost toward the distinguished end, and the corresponding component of $S$ is a half-catenoid.
