Bennett Chow, Peng Lu, Bo Yang
We show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler--Ricci shrinkers by Feldman, Ilmanen, and Knopf exhibit that this result is sharp.
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For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.
Noncompact self-shrinkers for mean curvature flow with arbitrary genus
本文利用 min-max 方法严格证明了 Ilmanen 关于任意亏格非紧自收缩子存在的猜想，并在大亏格情形下证实其恰有一个渐近锥形端，同时给出奇数亏格双端情形的数值证据。
We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.
An unknottedness result for noncompact self shrinkers
该研究将平均曲率流紧自收缩子的非打结定理推广至具一个渐近锥端的非紧自收缩子，论证运用平均曲率流，为相关拓扑分类研究提供支撑。
In this paper we consider a perturbation of the Ricci solitons equation proposed in \cite{jpb1} and studied in \cite{CaMa} and we classify noncompact gradient shrinkers with bounded nonnegative sectional curvature.
We prove a local graphical theorem for two-dimensional self-shrinkers away from the origin. As applications, we study the asymptotic behavior of noncompact self-shrinkers with finite genus. Also, we show uniform boundedness on the second fundamental form of two-dimensional noncompact self-shrinkers with bounded mean curvature and uniform locally finite genus.
Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
本文证明了 Kähler–Ricci 收缩子的标量曲率仅依赖维数的一致界，并建立了非紧收缩子无穷远处唯一切锥与其极化 Fano 纤维化基的同胚关系。
We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.
In this short note we observe that a recent result of C.-W. Chen meshes well with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L. Ni, and S.-J. Zhang to give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci solitons to have positive asymptotic volume ratio.
Self-shrinkers of the mean curvature flow in arbitrary codimension
对于维数大于1的超曲面，Huisken证明了具有正标量中曲率的中曲率流的紧致自收缩体是球面。我们将证明以下扩展：一个紧致的自相似的解决方案在任意余维和维数大于1是球形的，即包含在一个球，当且仅当中曲率向量\be H \ee是非零的和主法线\be ν \ee是平行的正常丛。我们还给出了这类完全非紧自收缩子的一个分类。
Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity
本文研究闵可夫斯基空间中非紧致类空严格凸超曲面的逆σ_k曲率流，证明其存在整体解且归一化后收敛到对应唯一自收缩子，丰富了非紧致情形下自收缩子的相关理论。
We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.
In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.
$κ$-solutions with the round cylinder as an asymptotic shrinker
研究显示，n≥4维Ricci流$κ$-解，其渐近收缩Ricci溶菌为圆柱体，必须均匀PIC。结合前期分类结果，此类非紧解要么为圆柱体收缩解，要么为Bryant稳态解；紧解则为Perelman古典型解。
A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers
本文针对非紧Kähler-Ricci收缩子证明了Matsushima定理、双全纯唯一性及极化Fano纤维化的K-多重稳定性。
Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities
该论文作为平均曲率流一般动力学系列的第二篇，针对首次奇点为渐近锥形收缩子的平均曲率流初始扰动展开研究，引入费曼-卡茨公式和非紧不变锥方法，证明一般初始扰动可使重标流避免锥形奇点。
Four-dimensional shrinkers with nonnegative Ricci curvature
本文通过Ricci曲率特征值演化渐近分析、带边界Gauss-Bonnet-Chern公式和分部积分，在截面曲率或数量曲率等几何条件下，将具有非负Ricci曲率的四维收缩孤立子分类为 R×S^3 或 R^2×S^2。
We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.
Quasilinear Liouville equation on manifolds with nonnegative Ricci curvature
研究非负Ricci曲率完备非紧Riemann流形上n-Laplace拟线性Liouville方程的刚性与分类，证明在尖锐对数下界条件下流形等距于欧氏空间且解为标准气泡，并构造非平坦流形上的解表明假设的尖锐性。
Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature
本文研究四维完备梯度收缩里奇孤立子在半正迷向曲率条件下的曲率估计与分类，给出非紧情形的二次曲率下界，并利用强极值原理完成相应分类。
A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature
本文研究了具有常标量曲率的收缩梯度Ricci溶剂的刚性性质，证明了其在特定条件下可视为有限商空间，扩展了Cheng-Zhou的结果到高维情况。
Complete Kähler manifolds with nonnegative Ricci curvature
本文研究非负Ricci曲率的完备Kähler流形，证明非负全纯双截曲率下流形的数量曲率积分渐近极限存在且估计最优，还发现Ricci平坦Kähler度量的强刚性，即单个无穷远切锥Ricci平坦则流形整体Ricci平坦且切锥唯一。
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