https://neckpinch.com/authors/siao-hao-guo/New arXiv papers by Siao-Hao Guo, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1909.06601/arxiv:1909.06601Sat, 14 Sep 2019 00:00:00 GMTSiao-Hao Guo — If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow.Mean curvature flowRicci solitonsSelf-shrinkers & translatorsStability & uniquenessLong-time existence & convergence https://neckpinch.com/papers/1807.02922/arxiv:1807.02922Mon, 09 Jul 2018 00:00:00 GMTSiao-Hao Guo — Given a mean curvature flow of compact, embedded C² surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.Mean curvature flow3-manifoldsSurfacesBoundaries & non-compactFree boundary & capillary https://neckpinch.com/papers/1701.01835/arxiv:1701.01835Sat, 07 Jan 2017 00:00:00 GMTSiao-Hao Guo, Natasa Sesum — J.J.L. Velacuteazquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C⁰ sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.Mean curvature flowSingularitiesMinimal & CMC surfaces https://neckpinch.com/papers/1611.03933/arxiv:1611.03933Sat, 12 Nov 2016 00:00:00 GMTSiao-Hao Guo — Given a smooth, symmetric, homogeneous of degree one function f(λ₁,⋯, λₙ) satisfying ∂ᵢf>0 for all i=1,⋯, n, and a rotationally symmetric cone 𝓒 in ℝⁿ⁺¹, we show that there is a f self-shrinker (i.e. a hypersurface Σ in ℝⁿ⁺¹ which satisfies f(κ₁,⋯, κₙ)+1/2X· N=0, where X is the position vector, N is the unit normal vector, and κ₁,⋯, κₙ are principal curvatures of Σ) that is asymptotic to 𝓒 at infinity.Other curvature flowsSelf-shrinkers & translatorsWeak & singular flowsHomogeneous spacesRotational symmetry & warped products https://neckpinch.com/papers/1604.08577/arxiv:1604.08577Thu, 28 Apr 2016 00:00:00 GMTSiao-Hao Guo — Given a smooth, symmetric, homogeneous of degree one function f=f(λ₁,⋯, λₙ) satisfying ∂ᵢf>0 for all i=1,⋯, n, and an oriented, properly embedded smooth cone 𝓒ⁿ in ℝⁿ⁺¹, we show that under some suitable conditions on f and the covariant derivatives of the second fundamental form of 𝓒, there is at most one f self-shrinker (i.e. an oriented hypersurface Σⁿ in ℝⁿ⁺¹ for which f(κ₁,⋯, κₙ)+1/2X· N=0 holds, where X is the position vector, N is the unit normal vector, and κ₁,⋯, κₙ are principal curvatures of Σ) that is asymptotic to the given cone 𝓒 at infinity.Other curvature flowsSelf-shrinkers & translatorsHomogeneous spacesStability & uniqueness