https://neckpinch.com/authors/junqi-lai/New arXiv papers by Junqi Lai, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2609.01346/arxiv:2609.01346Tue, 01 Sep 2026 00:00:00 GMTHaoxuan Cheng, Junqi Lai, Guoxin Wei — For every pair of integers 3≤ k<n with k odd, we construct a compact embedded rotational torus in ℝⁿ⁺¹ whose homothetic dilations satisfy the unnormalised σₖ-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity C^(1,1/k) and Sobolev regularity W^(2,p) for every 1≤ p<k/(k-1). Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation ⟨ X,ν⟩=−σₖ, where X is the position vector and ν is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical C² rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.Other curvature flowsRotational symmetry & warped products https://neckpinch.com/papers/2406.11123/arxiv:2406.11123Mon, 17 Jun 2024 00:00:00 GMTQing-Ming Cheng, Junqi Lai, Guoxin Wei — In the paper, we construct, for λ>0, complete embedded and non-convex λ-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that λ-hypersurfaces share a common conclusion on the planar domain conjecture even if the planar domain conjecture of T. Ilmanen for self-shrinkers of mean curvature flow are solved by Brendle affirmatively. Furthermore, for a fixed λ<0 which may have small |λ|, we can construct two compact embedded λ-hypersurfaces which are diffeomorphic to S¹× Sⁿ⁻¹, but they are not isometric to each other.Mean curvature flowSelf-shrinkers & translators