This paper concerns the evolution of a closed hypersurface of dimension n(≥ 2) in the Euclidean space ℝn+1 under a mixed volume preserving flow. The speed equals a power β(≥ 1) of homogeneous curvature functions of degree one and either convex or concave plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface.

中文翻译：

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一阶齐次曲率函数的混合体积保持流

这篇论文是关于一个封闭超曲面的演化n(≥ 2)在欧几里得空间ℝn+1在混合保容流下。速度等于功率β(≥ 1)一阶的齐次曲率函数和凸或凹加上混合体积保持项，包括平均曲率和高斯曲率的幂的情况。主要结果是，如果初始超曲面满足合适的收缩条件，则始终存在唯一的、平滑的流动解，并且演化的超曲面以指数方式收敛到一个圆形球体，包含与初始超曲面相同的混合体积。