In this paper, we consider the contracting curvature flows of smooth closed surfaces in 3-dimensional hyperbolic space and in 3-dimensional sphere. In the hyperbolic case, we show that if the initial surface \(M_0\) has positive scalar curvature, then along the flow by a positive power \(\alpha \) of the mean curvature H, the evolving surface \(M_t\) has positive scalar curvature for \(t>0\). By assuming \(\alpha \in [1,4]\), we can further prove that \(M_t\) contracts a point in finite time and become spherical as the final time is approached. We also show the same conclusion for the flows by powers of scalar curvature and by powers of Gauss curvature provided that the power \(\alpha \in [1/2,1]\). In the sphere case, we show that the flow by a positive power \(\alpha \) of mean curvature contracts strictly convex surface in \(\mathbb {S}^3\) to a round point in finite time if \(\alpha \in [1,5]\). The same conclusion also holds for the flow by powers of Gauss curvature provided that the power \(\alpha \in [1/2,1]\).

在本文中，我们考虑了光滑封闭表面在3维双曲空间和3维球面中的收缩曲率流。在双曲情况下，我们表明，如果初始曲面\（M_0 \）具有正的标量曲率，则沿流以平均曲率H的正幂\（\ alpha \）流动，演化的曲面\（M_t \）具有\（t> 0 \）的正标量曲率。通过假设\（\ alpha [in，1,4] \），我们可以进一步证明\（M_t \）在有限时间内收缩一点，并随着接近最终时间而变成球形。我们还给出了由标量曲率的幂和高斯曲率的幂组成的流的相同结论，条件是幂\（\ alpha \ in [1 / 2,1] \）。在球面情况下，我们证明，如果\（ \\\，则在有限时间内，平均曲率的正幂\（\ alpha \）的流在\（\ mathbb {S} ^ 3 \）中的严格凸表面收缩到一个圆点alpha \ in [1,5] \）。假设功率\（\ alpha \ in [1 / 2,1] \），则同样的结论也适用于高斯曲率的幂流。