https://neckpinch.com/authors/changyu-ren/New arXiv papers by Changyu Ren, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/2007.04493/arxiv:2007.04493Thu, 09 Jul 2020 00:00:00 GMTChangyu Ren, Zhizhang Wang, Ling Xiao — We prove three results in this paper. First, we prove for a wide class of functions φ∈ C²(Sⁿ⁻¹) and ψ(X, ν)∈ C²(ℝⁿ⁺¹×ℍⁿ), there exists a unique, entire, strictly convex, spacelike hypersurface Mᵤ satisfying σₖ(κ[Mᵤ])=ψ(X, ν) and u(x)→ |x|+φ(x/(|x|)) as |x|→∞. Second, when k=n-1, n-2, we show the existence and uniqueness of entire, k-convex, spacelike hypersurface Mᵤ satisfying σₖ(κ[Mᵤ])=ψ(x, u(x)) and u(x)→ |x|+φ(x/(|x|)) as |x|→∞. Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons Mᵤ with prescribed asymptotic behavior at infinity for σₖ curvature flow equations. Moreover, we prove that the downward translating solitons Mᵤ have bounded principal curvatures.Other curvature flowsSelf-shrinkers & translatorsStability & uniquenessPhysics