Haozhao Li, Yu Li, Bing Wang
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.
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We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.
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.
Compactness and Rigidity of Complete Kähler Ricci Shrinkers
本文借助极化法诺纤维化结构，探究完备凯勒里奇收缩子的紧性、刚性及相关几何性质。
Uniqueness of Kähler Ricci shrinkers on toric orbifolds
该研究证明了环轨形上Kähler Ricci收缩子的唯一性，将相关结果从环流形拓展到了环轨形范畴。
In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.
In this paper, we prove that the round cylinders are rigid in the space of Ricci shrinkers. Namely, any Ricci shrinker that is sufficiently close to $S^{n-1}\times \mathbb R$ in the pointed-Gromov-Hausdorff topology must itself be isometric to $S^{n-1}\times \mathbb R$.
This paper is the sequel to our study of heat kernels on Ricci shrinkers in \cite{LW20}. In this paper, we improve many estimates in \cite{LW20} and extend the recent progress of Bamler \cite{Bam20a}. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of $\IF$-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.
In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties.
Ends of (singular) Ricci shrinkers
本文研究了光滑和奇异Ricci收缩子的终点数量，特别探讨了渐近锥形结构的应用，证明极限Ricci收缩子中无法形成新的锥形终点。
On Kähler Ricci shrinker surfaces
该研究证明 Kähler Ricci 收缩曲面的截面曲率有界，并结合既有结果完成对此类曲面的完整分类。
Rigidity and ε-regularity theorems of Ricci shrinkers
本文研究Ricci收缩孤子的刚性与ε-正则性，证明渐近体积比和局部体积的刚性，改进局部熵与曲率的ε-正则性定理，并据此刻画无穷远处结构，给出曲率满足自然积分条件时渐近于锥的结论。
In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estimate, the no-local-collapsing theorem, the pseudo-locality theorem and the strong maximum principle for curvature tensors, are essentially improved for Ricci flows induced by Ricci shrinkers. Our results provide many necessary tools to analyze short time singularities of the Ricci…
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。
For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.
In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions.
Rigidity and gap theorems for Ricci shrinkers
本文证明了Ricci收缩体的局部Ricci曲率和ν-熵间隙定理，扩展了Munteanu-Wang和作者与Ma之前的成果，并提供了一种局部去除Ricci流I型奇点的标准。
Non-collapsing of Ricci shrinkers with bounded curvature
为具有限第二同伦群和一致曲率界的单连通Ricci收缩子建立一致熵界，并将非塌陷结果扩展至满足Bakry-Émery条件的光滑度量测度空间。
Rigidity of complex projective spaces in Ricci shrinkers
本文证明了在Gromov-Hausdorff意义下充分接近复射影空间的Ricci收缩子必等距于该空间，建立了刚性定理。
Some uniform estimates for Kähler--Ricci Shrinkers
本文在无曲率假设下，对Kähler-Ricci收缩子建立了熵、渐近体积比、标量曲率的一致下界，丰富了Kähler-Ricci收缩子的相关估计理论。
In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new feature is that no curvature upper bound is assumed.
Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers
研究非塌缩凯勒 - 里奇流有限时间奇点及里奇收缩子无穷远处几何，证明凯勒 - 里奇流时间切片收敛等结论，还将其与法诺纤维化联系，应用于证明特定收缩子无穷远处切空间唯一性。
In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.
Heat kernel on Ricci shrinker metric measure spaces
本文系统研究Ricci收缩度量测度空间上的热核，建立基本估计，阐明其与Ricci流时空热核的等价性，并推广Nash熵与约化距离，进而证明梯度估计蕴含曲率衰减。
In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact can be removed. The key ingredient is a recent estimate of Cheeger-Naber arXiv:1406.6534.
We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.
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.
On the rigidity of Ricci shrinkers
本文通过定量刚性刻画、混合阶刚性不等式及收缩扩张方法，证明了广义柱面在Ricci收缩子空间中的刚性，并由此得到一般紧致Ricci流在切流为广义柱面时的切流唯一性。
Geometric Structure of Ends of Ricci Shrinkers
本文在无全局曲率假设下研究 Ricci 收缩子端部的几何结构，证明标量曲率满足 Type I 界的点处其 F-极限分裂出直线，并扩展了 F-收敛理论。
Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers
研究梯度瑞奇收缩体序列在熵和局部能量界限制下的气泡树收敛，推导局部能量恒等式及欧拉特征恒等式，证明局部微分同胚有限性。
An eigenvalue estimate for self-shrinkers in a Ricci shirinker
本文研究Ricci收缩孤立子中超曲面上漂移拉普拉斯算子的谱性质，证明了有界加权平均曲率浸入超曲面的谱离散性，并给出嵌入f-极小超曲面第一非零特征值的下界，推广了Choi-Wang等经典估计。
Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker
该研究针对非紧梯度收缩Kähler-Ricci孤立子，在标量曲率满足温和假设的条件下，证明其上多项式增长全纯函数环OP(X)有限生成，部分证实了Munteanu和Wang的相关猜想。
Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers
本文针对Ricci收缩流形上的Schrödinger算子，利用收缩熵和势函数给出了最低特征值的下界，并推广到完备黎曼流形及带漂移的Schrödinger算子。
On second variation of Perelman's Ricci shrinker entropy
本文给出Perelman的ν-熵第二变分公式的详细证明，纠正了已有文献中稳定性算子的错误，并通过相关Lichnerowicz型算子的最小特征值及其重数得到线性稳定收缩子的必要条件。
Kähler-Ricci shrinkers and Fano fibrations
本文用双有理代数几何证明Kähler-Ricci收缩孤立子是拟射影簇，并提出将其存在性与极化Fano纤维化K-稳定性及奇点切流相联系的统一猜想。
Propagation of symmetries for Ricci shrinkers
该研究针对梯度收缩里奇孤立子的对称性问题，证明其单尺度近似对称性可传播至更大尺度，为收缩子原理提供了实例支撑。
In this paper we have found some commutator formulas between $div_{f},Δ_{f,L},div_{f}^\dagger,Δ_{f}$ on closed orientable $GRS^+$ metrics (Gradient Ricci Shrinker), then with them we have generalized a Theorem of Cao and Zhu about necessary condition for linear stability of a $GRS^+$.
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.
We prove a local gap theorem for Ricci shrinkers, which states that if the local $μ$-functional at scale $1$ on a large ball centered at the minimum point of the potential function is close enough to $0$, then the shrinker must be the flat gaussian shrinker. In relation to our result, Yokota [Yo09,Yo12] proved the same result assuming the global $μ$-functional to be close enough to $0$. Our result shows an aspect of how the local geometry of a shrinker controls the global geometry, which is also discussed in [LW19,LW20,LW21].
Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
本文证明任意余维数中满足 Ricci 下界条件的完备自收缩子必为线性子空间或圆球，并给出余维数二时的尖锐刚性定理。
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。
The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers
本文证明完备梯度Ricci收缩子具有大规模正则性，并由此推出其加权拉普拉斯算子满足经典Weyl定律。
K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers
本文证明了渐近锥形Kähler-里奇收缩者度量的存在性与K-极稳定性之间的等价关系。
The Wasserstein distance for Ricci shrinkers
本文研究 Ricci 收缩子切空间中指数映射诱导测度与 Gaussian 测度的 Wasserstein 距离，证明其关于欧氏度量的上界估计及相应刚性结论。
We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.
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.
Linear stability of the blowdown Ricci shrinker in 4D
本文证明了四维吹down Ricci收缩器在特定意义下的线性稳定性，并揭示了Ricci流在四维中的拓扑行为新特性。
Spectral bounds for $f$-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers
研究完备梯度 Ricci 收缩子上$f$-拉普拉斯算子谱，依体积增长率建特征值上下界，方法可推广到向量丛，还获贝蒂数明确上界。
In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking Kähler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li about the non-existence of compact self-expanders in a Euclidean space.
Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
该研究证明在闭合流形上存在一个里齐流解，其在有限时间形成基于渐近圆锥收缩子流形的奇异性和模型。
Singular Kähler--Ricci shrinkers and polarized Fano fibrations
本文证明任何作为 Kähler-Ricci 流非塌缩极限的奇异 Kähler-Ricci 收缩子具有局部代数复解析簇结构，并推导出单连通、唯一端、唯一切触锥等几何性质。
$κ$-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。
Shrinkers with curvature pinching conditions are compact
本文通过建立Ricci曲率正性与渐近非负截面曲率两类曲率夹挤条件，证明了相应收缩孤立子的紧致性，并给出相关分类结果。
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。
Dimension estimate and existence of holomorphic sections with polynomial growth on gradient Kähler Ricci shrinkers
本文证明了有界曲率梯度Kähler Ricci收缩子上多项式增长全纯函数维数的幂函数上界，并推广到$(p,0)$-形式和$K_M^{-q}$截面；对渐近锥形情形证明了充分大$q$时截面的存在性，并由此构造出全纯嵌入的Kodaira映射。
Four-dimensional shrinkers with pinched curvatures
本文针对四维收缩子，在不同曲率挤压条件下完成分类，并建立Hitchin-Thorpe型不等式，丰富了收缩子的几何研究成果。
Convergence of Ricci flow and long-time existence of Harmonic map heat flow
本文建立了渐近紧致可积shrinker的ancient Ricci flow及相应奇点情形下harmonic map heat flow的长时间存在性，提供全局parabolic gauge并证明切流唯一性，同时分类了ancient flows并得出奇点最优收敛速率。
The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers
研究完整梯度收缩Ricci流形上的多项式增长全纯函数和形式，通过f-拉普拉斯谱数据证明其空间维数有限，并获得线性增长全纯函数的精确维数估计。
Semi-Continuity of the Morse Index for Ricci Shrinkers
本文证明梯度 Ricci 收缩子序列在泡树收敛下 Morse 指标的半连续性，并改进方法以处理非紧情形，给出渐近锥形收缩子指标下界条件。
Backward propagation of warped product structures and asymptotically conical shrinkers
本文研究Ricci流中扭曲积结构的向后传播，并证明渐近锥形梯度收缩孤子若渐近于Einstein流形乘积锥，则其自身为多重扭曲积。
In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a bound can be obtained from an eigenvalue estimate for a twisted Laplacian, called the Witten-Laplacian. In the second part we discuss self-similar solutions to the mean curvature flow on cone manifolds. Many results have been obtained for solutions in $\bfR^n$ or $\bfC^n$. We see tha…
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