https://neckpinch.com/authors/jingyi-chen/New arXiv papers by Jingyi Chen, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1604.07563/arxiv:1604.07563Tue, 26 Apr 2016 00:00:00 GMTJingyi Chen, John Man Shun Ma — We study geometric properties of the Lagrangian self-shrinking tori in ℝ⁴. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in. When the area bound is small, we show that any Lagrangian self-shrinking torus in ℝ⁴ with small area is embedded with uniform curvature estimates, and the space of such tori is compact. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in ℝ⁴, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in ℝ³ in.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesLimit spaces & RCD