Dongsheng Li, Yingfeng Xu, Yu Yuan
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.
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Rigidity of complete entire self-shrinking solutions to Kahler-Ricci flow
证明了Kahler-里奇流在复欧几里得空间上的每一个完全整自缩解都必须由一个二次势生成.
We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.
Rigidity of Entire self-shrinking solutions to curvature flows
我们证明了：（a）具有欧几里德度量的${\mathbb C}^{m}$中的拉格朗日中曲率流的任何整图自缩解是平坦的;（b）具有伪欧几里德度量的${\mathbb C}^{m}$中的拉格朗日中曲率流的任何类空整图自缩解是平坦的，如果位势的Hessian在下面二次有界;以及（c）对于Kähler里奇流，（b）的Hermitian对应。
Geometric inequalities and rigidity of gradient shrinking Ricci solitons
该研究证明完备梯度收缩 Ricci 孤立子上六类几何不等式与热核估计等价存在，并给出紧致情形下的积分间隙刚性结果。
Rigidity of asymptotically conical shrinking gradient Ricci solitons
证明了如果两个梯度Ricci孤子沿各自的某个端点渐近于同一个正则锥，则孤子度量在这些端点的无穷远邻域上一定是等距的。我们的定理对度量的行为没有任何限制，特别是不需要它们的测地线完备性。作为应用，我们证明了R^n上唯一渐近于旋转对称锥的完全连通梯度收缩Ricci孤子是高斯孤子。
Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas
研究四维梯度收缩黎曼流形在 Bach-like 张量消失时的刚性性质，并探讨其与二次曲率函数的变分关系。
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。
A Bernstein Theorem for the Self-Shrinking $J$-Equation and Some Generalizations
本文证明 $\mathbb{C}^n$ 上自收缩 $J$-方程的每个整体光滑多重次调和解必为二次多项式，去除了复 Hessian 的渐近下界假设，并推广到一大类完全非线性椭圆算子。
The Curvature of Gradient Ricci Solitons
研究了梯度收缩Ricci孤子曲率的积分和点态界。作为应用，我们讨论梯度收缩的间隙和紧性结果。
Applications of P-functions to Fully Nonlinear Elliptic equations: Gradient Estimates and Rigidity Results
本文引入全非线性方程的P-函数概念，建立一般准则，获得梯度界、De Giorgi型性质与刚性结果，推广Modica不等式，并证明Harnack型不等式及高阶方程的Liouville定理。
This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in the three-dimensional Euclidean space $\mathbb{R}^3.$ We prove that an immersed self-shrinker with finite $L$-index must be proper and of finite topology. As one of consequences, there is no stable two-dimensional self-shrinker in $\mathbb{R}^3$ without assuming properness. We conclude the paper by giving an affirmative answer to a question of Mantegazza.
We propose a natural definition of the weighted $σ_k$-curvature for a manifold with density; i.e.\ a triple $(M^n,g,e^{-ϕ}\mathrm{dvol})$. This definition is intended to capture the key properties of the $σ_k$-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted $σ_k$-curvature functionals when the weighted $σ_k$-curvature is variational. Second, we characterize th…
Rigidity of weighted manifolds via classification results for semilinear equations
本文通过半线性方程的分类结果，研究加权流形的刚性，并证明正解的存在性与流形的刚性及权重的平凡性之间的关系。
We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinear curvature flows of hypersurfaces.
Nonlinear flows and rigidity results on compact manifolds
本文研究光滑紧致连通无边界黎曼流形上某些椭圆型偏微分方程的刚性结果及相关的Sobolev型插值不等式。刚性是指当参数在一定范围内时，偏微分方程除了常数解外没有其它解。该参数可作为相应插值不等式中最佳常数的估计。我们的方法依赖于多孔介质/快速扩散型的非线性流动，它给出了一个明确的解释，在早期的作品中所做的指数的技术选择。我们还建立了两个积分的刚性标准，提高了已知的，逐点条件，并举行一般流形上的曲率没有积极性条件。利用该流，我们还可以讨论插值不等式中相应常数的最优性。
Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows
该研究证明了五维定常Navier-Stokes方程的临界Morrey刚性定理，推导了五维可去奇点准则，还通过有限能量论证证明了四维对应的三次Morrey刚性定理。
Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem
本文证明在光滑紧致域上，若一个黎曼度量与欧几里得度量具有相同的边界度量和Dirichlet-to-Neumann映射且足够接近，则它们通过固定边界的微分同胚等距，即欧几里得度量具有局部刚性。
Rigidity of a $Q$-curvature-type problem on $\mathbb{S}^N$ in every dimension $N\geq3$
该研究证明了任意维数N≥3时，球面S^N上满足α≥1/2且α≠1的Q-曲率型问题的所有解均为常数，其核心方法为谱间隙估计与归一化密度的点态估计。
Smoothing polyhedral spaces via Ricci flow
本文证明了Petrunin光滑化猜想：任意紧致无边界非负Alexandrov曲率的多面体空间是光滑黎曼轨形空间的Gromov-Hausdorff极限，通过新Ricci流短时间存在理论实现，并给出积分曲率缺陷的刚性应用。
Rigidity of ancient ovals in higher dimensional mean curvature flow
本文研究高维平均曲率流中$k$-椭圆的分类与刚性，证明其具有特定对称性且由谱比参数唯一确定，通过新测试张量和局部参数化方法解决领圈区域估计等障碍，建立谱稳定性定理。
Universal Spacelikeness Estimates and Liouville Rigidity for Lorentzian $σ_k$ Curvature Equations
研究Lorentz-Minkowski空间中满足$\sigma_k$方程的类空图，建立高度与Lorentz因子的先验界，并在一定参数范围内证明解恒为零的Liouville刚性定理。
On the classification of ancient solutions to curvature flows on the sphere
本文研究单位球面上曲率流的演化，提出“准古典”解概念，并证明满足平均曲率向后时间一致界的凸准古典解必为稳态解或收缩测地球，主要运用极值原理与 Alexandrov 反射等几何工具。
Exact mean-covariance dynamics of the Weber field in the stochastic Lagrangian representation of the 3D Navier-Stokes equations
该研究推导三维Navier-Stokes方程随机拉格朗日表示中韦伯场的均值-协方差动力学方程，获能量平衡等结果，指出正则性相关障碍并提出开放问题。
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