The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures \(f(k_1,\ldots ,k_{n-1})\) satisfying suitable conditions. In this paper, we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator f is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is optimally quantified in terms of the oscillation of the curvature function f. Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.

亚历山德罗夫肥皂泡定理断言，距离球是具有恒定平均曲率的空间形式中唯一嵌入的封闭连接超表面。该定理可以扩展为满足合适条件的主曲率\（f（k_1，\ ldots，k_ {n-1}）\）的更一般的函数。在本文中，当曲率算子f接近常数时，我们对空间形式的Alexandrov肥皂泡定理给出了接近单个球的精确定量估计。在防止起泡的假设下，根据曲率函数f的振荡，可以最佳地量化与单个球体的接近度。我们的方法为以空间形式移动平面方法的定量研究提供了统一的图片。