Shunzi Guo

This paper concerns the evolution of a closed hypersurface of dimension $n(\ge 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $\beta (\ge 1)$ of homogeneous, either convex or concave, curvature functions of degree one 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. This result covers and generalises the previous results for convex hypersurfaces in the Euclidean space by McCoy and Cabezas-Rivas and Sinestrari to more general curvature flows for convex hypersurfaces with similar curvature pinching condition.

This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This result covers and generalises the corresponding result of Schulze for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.