https://neckpinch.com/authors/liuyang-zhang/New arXiv papers by Liuyang Zhang, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2508.07900/arxiv:2508.07900Mon, 11 Aug 2025 00:00:00 GMTShu-Cheng Chang, Hongbing Qiu, Liuyang Zhang — We study the self-shrinkers of the Legendrian mean curvature flow in the contact Euclidean space ℝ²ⁿ⁺¹, that is, the Legendrian submanifolds which move under the flow by the contact dilations (x,y,z)↦(λx,λy,λ²z), and we explain why it is the generator of these dilations, and not the Euclidean position vector, which appears in the self-shrinker equation. Our main result is that a complete properly immersed Legendrian self-shrinker with bounded Legendrian angle is a Legendrian n-plane. On a Legendrian self-shrinker the Legendrian angle is a constant multiple of the coordinate z, so that this is a statement about the position of the submanifold, and it has two consequences. A complete properly immersed Legendrian self-shrinker which lies in a slab |z|≤ C is a Legendrian n-plane, and in particular there are no closed Legendrian self-shrinkers in ℝ²ⁿ⁺¹; this has no counterpart for Lagrangian self-shrinkers in ℂⁿ, among which there are closed ones. And the potential of an entire smooth Legendrian self-shrinking graph is a quadratic polynomial, without any further assumption. The proof uses a volume growth estimate for Legendrian self-shrinkers with respect to a Gaussian weight which is built from the horizontal part of the generator of the dilations.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsStability & uniqueness https://neckpinch.com/papers/2312.15996/arxiv:2312.15996Tue, 26 Dec 2023 00:00:00 GMTShu-Cheng Chang, Chin-Tung Wu, Liuyang Zhang — In this article, we recapture the Smale conjecture on a Sasakian 3-sphere via the Legendrian mean curvature flow. More precisely, we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian 3-spheres to an isometry via the Legendrian mean curvature flow on the Legendrian graph in S²× S³. By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in S²× S³. Finally, we will address the rigidity theorem of 2-dimensional Legendrian self-shrinkers in ℝ⁵. We are able to reconstruct the Harvey-Lawson special Lagrangian cone in ℂ³ from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesSmale conjecture & diffeomorphismsStability & uniquenessSasakian & contact geometry