Max Hallgren
In this short paper, we show there do not exist three-dimensional noncompact $κ$-solutions of Ricci flow that have positive curvature and satisfy a Type-I bound. This represents progress towards the proof of Perelman's conjecture that the only complete noncompact three-dimensional $κ$-solution with positive curvature is the Bryant soliton.
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Let $M$ be a compact Kähler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$. Then $M-E$ cannot admit any complete Kähler--Einstein metric if blow-down of $E$ is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs.
The goal of this article is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain pde estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.
Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds
研究了非正曲率完备非紧黎曼流形上的共形数量曲率方程。得到了数量曲率方程上解的一些估计和性质，并得到了数量曲率方程完全解的不存在性结果.
In this article, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three-dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.
Nonexistence of Finite-dimensional Quantizations of a Noncompact Symplectic Manifold
本文证明了在非紧辛流形M上不存在一个“观测值的基本代数”B的忠实的斜厄米矩阵有限维表示。因此，包含B的泊松代数C^ \infty（M）的任何李子代数都不存在有限维量子化。
On the asymptotic scalar curvature ratio of complete Type I-like ancient solutions to the Ricci flow on non-compact 3-manifolds
本文的主要结果是：给定任意常数C，存在$(ε,k,L)$使得如果一个具有有界正截面曲率的完备、可定向、非紧奇维流形包含$(ε,k,L)$ -颈，则渐近数量曲率比大于或等于C.作为应用，我们证明了可定向三维流形上完全非紧古I型解与具有有界正截面曲率的里奇流的渐近数量曲率比为无穷大.
We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.
Nonexistence of solutions to $Δ_pu+Δ_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds
本文通过测试函数方法，在测地完备非紧黎曼流形上证明了含梯度项的不等式解的Liouville型定理，推广了已有结果，并讨论了高阶项在s<0时的影响。
Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image
本文通过进行精细的曲率估计，在各种高斯像限制条件下，证明了高余维平均曲率流完备非紧古老解的更好刚性定理。
A rigidity theorem of ancient solutions to the mean curvature flow in codimension one
该研究针对余维一平均曲率流的完备非紧古代解，通过对第二基本形式做逐点估计，证明了相关刚性定理，并推导得到最优增长条件。
It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $κ$-solution. We prove that the every noncompact ancient $κ$-solution in dimension $3$ is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.
Type II ancient solutions to the Ricci flow on surfaces
曲面上的里奇流的II型（古）解还没有分类.它是证明Rosenau解决方案和雪茄是唯一的解决方案，模缩放。本文主要研究非紧曲面上第二类古解的后向极限和无穷远点的周长。
In [4], we proved that every noncompact ancient $κ$-solution to the Ricci flow in dimension $3$ is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that the same method implies that compact ancient $κ$-solutions are rotationally symmetric. In this note we provide the details of this argument.
A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $κ$-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $\mathbb{R}^3$ which is strictly convex and noncollapsed.
Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions
针对高维欧氏空间中满足严格凸、一致2-凸、非坍塌条件的平均曲率流非紧古老解，证明其必为旋转对称平移孤立子，完善了平均曲率流古老解的分类理论。
A nonexistence result for rotating mean curvature flows in $\mathbb{R}^{4}$
针对平均曲率流中可能作为奇点模型的旋转古老流，本文利用切流轴唯一性无法约束内部旋转这一观察，证明在 $\mathbb{R}^4$ 的古老非坍缩流中不存在此类旋转流。
We show that every $n$-dimensional, $κ$-noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for $n=4$ or $n\ge 12$ has weakly PIC$_2$ and bounded curvature. Combining this with earlier results, we prove that any such solution is isometric to either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton. Also, we classify all complex 2-dimensional, $κ$-noncollapsed, complete ancient solutions to the Kähler Ricci flow with weakly PIC.
Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type~I curvature growth. We show that a $κ$-noncollapsed Type~I ancient solution which is noncompact and has nonnegative sectional curvature necessarily splits at least one Euclidean f…
Revisiting ancient noncollapsed flows in $\mathbb{R}^3$
本文结合精细颈部定理与 Hamilton Harnack 不等式的刚性情形，为 Brendle-Choi 关于三维非紧古老非塌缩流的分类定理提供了直接确立自相似性的新证明。
Non-ancient solution of the Ricci flow
对任意具有非负有界全纯双截曲率的完备非紧$\ddot{a}$流形，本文给出了里奇流非古解的充要条件.
In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete $3$-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient -- a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.
Stable hypersurfaces with constant scalar curvature in Euclidean spaces
在欧几里得空间中得到了具有非负常数数量曲率的完备非紧稳定超曲面的不存在性结果。作为特例，我们证明了在具有零数量曲率$S_2$、非零Gauss-Kronecker曲率和有限全曲率的$\mathbb{R}^{4}$中不存在完备的非紧强稳定超曲面$M$（即$\int_M|A|^3<+\infty$）.
Liouville theorems for ancient solutions to the V-harmonic map heat flows II
本文利用精细梯度估计，在非负Bakry–Émery Ricci曲率与目标流形正曲率上界条件下，改进V-调和映射热流古老解的Liouville定理，并得到指数增长条件下拟调和映射的相应结果。
Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions
本文把古代 κ-解唯一性推广到高维，证明 n≥4 时非紧古代 κ-解仅为收缩圆柱、其商或 Bryant 孤子。
We generalize the famous result of Gromov and Lawson on the nonexistence of metric of positive scalar curvature on enlargeable manifolds to the case of foliations, without using index theorems on noncompact manifolds.
In this paper, we study $κ$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$. By making use of the properties of such ancient solutions, we generalize part one of Brendle \cite{brendle2018ancient} to higher dimensions, that is, every noncompact $κ$-noncollapsed rotationally symmetric ancient solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton.
On a paper of Daskalopoulos and Sesum
本文是Daskalopoulos和Sesum关于里奇流的任何具有有界正数量曲率和有限宽度的二维完全非紧古解必定是雪茄孤子的结果的一个方面的阐述。
A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$
本文排除了$\text{R}^4$中3维超曲面平均曲率流的非塌缩翼状奇点模型，证明古老非塌缩流的爆破极限非楔体，提出精细气泡片分析，为$\text{R}^4$古老非塌缩流分类奠定基础。
Rotational symmetry of ancient solutions to fully nonlinear curvature flows
本文对完全非线性曲率流的凸非坍缩一致二凸非紧古解进行分类，证明其仅可能为自相似收缩圆柱或旋转对称平移孤子，并由此给出紧致二凸解奇点爆破极限的完整分类。
A differential Harnack inequality for noncompact evolving hypersurfaces
本文证明非紧致凸超曲面在对称曲率函数速度下流动的微分Harnack不等式，推广Andrews的紧致结果，并适用于奇点爆破产生的古代解。
Ancient Solutions of the Affine Normal Flow
我们构造了超曲面的仿射法线流的非紧解，并证明了所有的古代解必须是椭球（收缩孤子）或抛物面（平移孤子）。我们还提供了一个新的证明，双曲仿射球的存在性渐近的凸锥的边界不包含线，这最初是由于Cheng-Yau。主要的技术是本地二阶导数估计的抛物Monge-Ampere方程建模的本安德鲁斯和Gutierrez-Huang，衰减估计的三次型下的仿射法线流由于本安德鲁斯，和一个超曲面障碍由于卡拉比。
Nonexistence of isoperimetric sets in spaces of positive curvature
该研究构造了正曲率黎曼流形及凸集中小体积下等周集不存在的例子，证明维度下界3为最优，并首次给出无等周集的非塌缩非负曲率空间。
Classification of ancient ovals in higher dimensional mean curvature flow
本文研究高维均曲率流中古老卵形的分类，通过不同的谱参数化方法，对$k$-卵形进行了独立分类，并与Bamler-Lai的突破性成果形成对比。
CMC hypersurface with finite index in hyperbolic space $\mathbb{H}^4$
本文利用μ-bubble方法证明了双曲空间$\u210D^4$及满足曲率条件的4维黎曼流形中，不存在特定的完备非紧常平均曲率/极小稳定超曲面，推进了几何测度论相关不存在性问题的研究。
On dimension reduction in the Kähler-Ricci flow
本文考虑具有非负二分曲率的Kähler-里奇流解的降维问题。当复维数为$n=2$时，证明了具有非负二分曲率的完全平移Kähler-Ricci孤子的最优降维定理.在有限性假设下，证明了非紧复流形上具有非负对分曲率的里奇流的完备古解的一般降维定理.
Generic Properties of Homogeneous Ricci Solitons
讨论了齐次Ricci孤子的几何性质。在证明了紧致均匀孤子和非紧致稳态均匀孤子的不存在性之后，我们集中研究了左不变Ricci孤子。证明了在幺模情形下，Ricci孤子方程在左不变向量场的集合中不存在解。证明了梯度型左不变孤子必是具有非平凡Euclidean de Rham因子的黎曼积。作为结果的应用，我们证明了任何广义度量Heisenberg李群都是非梯度的膨胀型左不变Ricci孤子.
$κ$-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。
We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the Euclidean space, it is known from Herrero and Pierre [20] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in $ L^1_{\mathrm{loc}} $. We establish here that such data still give rise to global solutions on general Riemannian manifolds. If, in addition, the radial Ricci curvature satisfies a suitable pointwise bound…
No compact split limit Ricci flow of type II from the blow-down
本文利用Perelman的$\mathcal L$-测地线理论，证明了在满足特定曲率条件的非紧$κ$-非坍缩稳态梯度Ricci孤立子中，由blow-down产生的余维一紧分裂古老解必为I型，并将此结论推广至任意维度。
Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set
本文研究紧集外曲率非负的完备非紧稳态Ricci孤立子，证明其缩并的分裂极限流的相关不存在性与共存性结论，并应用该结论给出一类孤立子等距于Bryant Ricci孤立子的判定条件。
Classification of ancient noncollapsed flows in $\mathbb{R}^4$
本文通过证明微分颈定理，完整分类了R^4中平均曲率流的非塌缩古代解，证明其必为经典例子或两类已知单参数族，并排除了奇异椭圆。
Pinching cones for positive isotropic curvature in dimensions seven and eight
本文针对七、八维Hamilton常微分方程构造横截不变捏紧锥，修复Chen构造在n=7、8的失效步骤，结合相关结果扩展正各向同性曲率流形及Ricci流古老解的分类定理。
Quasi-linear equation $Δ_pv+av^q=0$ on manifolds with integral bounded Ricci curvature and geometric applications
本文研究了在积分有界曲率的流形上 quasi-linear 方程 $Δ_pv+av^q=0$ 的非存在性结果与梯度估计，并给出了几何应用。
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