https://neckpinch.com/authors/jun-sun/New arXiv papers by Jun Sun, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.18064arxiv:2609.18064Wed, 16 Sep 2026 00:00:00 GMTLi Xiang, Sun Jun — A λ-translating soliton is a hypersurface in ℝⁿ⁺¹ satisfying H=⟨ T,ν⟩+λ; equivalently, it has constant weighted mean curvature with respect to the log-linear density e^(⟨ T,X⟩), and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed λ-translating soliton with λ>0 and inf_ΣH>λ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic λ-translating solitons (λgeqslant 0) with bounded gradient, and two rigidity theorems.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsStability & uniqueness https://arxiv.org/abs/2511.00435arxiv:2511.00435Sat, 01 Nov 2025 00:00:00 GMTYaoting Gui, Yuqiao Li, Jun Sun — In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.Mean curvature flowVolume-preserving & constrained flowsStability & uniquenessBoundaries & non-compactLong-time existence & convergenceGeometric inequalitiesMinimal & CMC surfaces https://arxiv.org/abs/2412.17024arxiv:2412.17024Sun, 22 Dec 2024 00:00:00 GMTYaoting Gui, Yuqiao Li, Jun Sun — This paper investigates the volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces. We demonstrate the long-time existence and exponential convergence of this flow with a coordinate sphere of large radius serving as the initial surface in the asymptotically flat end, which eventually converges to a constant harmonic mean curvature surface. We also establish that these surfaces form a foliation of the space outside a large ball. Finally, we utilize this foliation to define the center of mass, proving that it agrees with the center of mass defined by the ADM formulation of the initial data set.Mean curvature flowBoundaries & non-compactLong-time existence & convergencePhysics https://arxiv.org/abs/2412.15880arxiv:2412.15880Fri, 20 Dec 2024 00:00:00 GMTXiang Li, Yong Luo, Jun Sun — In this paper, we will prove some rigidity theorems for blow up limits to Type II singularities of Lagrangian mean curvature flow with zero Maslov class or almost calibrated Lagrangian mean curvature flows, especially for Lagrangian translating solitons in any dimension. These theorems generalized previous corresponding results from two dimensional case to arbitrarily dimensional case.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesSurfacesStability & uniqueness https://arxiv.org/abs/2410.17850arxiv:2410.17850Wed, 23 Oct 2024 00:00:00 GMTXiaoli Han, Jiayu Li, Jun Sun — It is known that there is no a Type I singularity for the Lagrangian mean curvature flow with zero Maslov class. In this paper, we study translating solitons which are important models of Type II singularities. A necessary condition for a blow-up limit arising at a Type II singularity of a Lagrangian mean curvature flow with zero Maslov class is provided. As an application, we try to understand the important open question proposed by Joyce-Lee-Tsui and Neves-Tian, whether the Lagrangian translating solitons constructed by Joyce-Lee-Tsui can be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularities https://arxiv.org/abs/2409.12810arxiv:2409.12810Thu, 19 Sep 2024 00:00:00 GMTYaoting Gui, Yuqiao Li, Jun Sun — We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is sufficiently small. Then we show that from sufficiently large initial coordinate sphere, the area preserving mean curvature flow exists for all time and converges exponentially fast to a constant mean curvature surface in 3-dimensional asymptotically Schwarzschild spaces. This provides a new approach to the existence of foliation established by Huisken and Yau. And also a uniqueness result followsMean curvature flowVolume-preserving & constrained flows3-manifoldsSphere theoremsStability & uniquenessLong-time existence & convergenceMinimal & CMC surfaces