AI 中文总结
该研究针对三维光滑度量测度空间中的亏格零紧致浸入曲面，引入新的加权平均曲率，证明满足特定条件时该类曲面若加权平均曲率为常数则必为全脐曲面，并将结果应用于对偶Christoffel-Minkowski问题的唯一性研究。
AI 中文摘要
我们考虑三维光滑度量测度空间$(M^3,g,e^{-f}\mathrm{d}V_{g})$中亏格为零的紧致浸入曲面，引入一种新的加权平均曲率，证明若满足与$M$的里奇曲率和权重函数$f$相关的条件，任何加权平均曲率为常数的此类曲面必为全脐曲面，还将该结果应用于对偶Christoffel-Minkowski问题的唯一性研究。
英文摘要
We consider compact immersed surfaces of genus zero in a three-dimensional smooth metric measure space \((M^3,g,e^{-f}\mathrm{d}V_{g})\). We introduce a new weighted mean curvature and prove that any such surface whose weighted mean curvature is constant must be totally umbilical, provided that a condition relating the Ricci curvature of $M$ and the weight function $f$ is satisfied. We also give an application of this result to the uniqueness of the dual Christoffel-Minkowski problem.
