https://neckpinch.com/authors/shunzi-guo/New arXiv papers by Shunzi Guo, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1610.08214/arxiv:1610.08214Wed, 26 Oct 2016 00:00:00 GMTShunzi Guo — 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, 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.Other curvature flowsVolume-preserving & constrained flowsCurvature pinchingHomogeneous spacesLong-time existence & convergence https://neckpinch.com/papers/1610.08209/arxiv:1610.08209Wed, 26 Oct 2016 00:00:00 GMTShunzi Guo — This paper concerns the evolution of a closed convex hypersurface in ℝⁿ⁺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.Mean curvature flow