Marcos Craizer, Ronaldo A. Garcia
In this paper we discuss the behavior of the curvature lines of a transversal eqüiaffine vector field along a surface in $3$-space at isolated umbilical points.
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Totally umbilical radical transversal lightlike hypersurfaces of Kähler-Norden manifolds of constant totally real sectional curvatures
本文研究了具有常全实截面曲率$\overline ν$和$\overline {\widetilde ν}$的Kähler-Norden流形$(\overline M,\overline J,\overline g,\overline {\widetilde g})$的半对称型全脐根横截类光超曲面$(M,g)$和$(M,\widetilde g)$的曲率性质（$g$和$\widetilde g$分别是Norden度量$\overline g$和$\overline {\widetilde g}$在$M$上的诱导度量）.我们得到了$\overline {\widetilde ν}$的一个条件（分别为. $\overline ν$是局部对称的、半对称的、Ricci半对称的和几乎爱因斯坦的。我们构造了一个全脐根横截类光超曲面的例子，它是局部对称的，半对称的，Ricci半对称的，几乎是爱因斯坦的。
Let $(M,g_M,\mathcal F)$ be a closed, connected Riemannian manifold with a Riemannian foliation $\mathcal F$ of nonzero constant transversal scalar curvature. When $M$ admits a transversal nonisometric conformal field, we find some generalized conditions that $\mathcal F$ is transversally isometric to the sphere.
Transversal Surfaces of Timelike Ruled Surfaces in Minkowski 3-Space
本文给出了类时直纹曲面的横截曲面的定义和刻画。研究了收缩曲线为测地线、渐近线或曲率曲率线等特殊情况。此外，我们还得到了类时直纹曲面的横截曲面的可展条件。
In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection the normals of the surfaces at the intersection point are linearly independent, while as in nontransversal intersection the normals of the surfaces at the intersection point are linearly dependent.
On complete constant mean curvature vertical multigraphs in E(κ,τ)
证明了在齐性空间E（κ，τ）中横截于垂直Killing向量场的任何具有常中曲率的完备曲面实际上是垂直图.因此，我们得到，在E（κ，τ）中具有常数中曲率H的任何可定向的、抛物的、完全的、浸入的曲面（不同于S^2 × R中的水平切片）要么是垂直柱，要么是垂直图（在这两种情况下，它都必须是4H^2+κ \leq 0）。
Equiaffine Darboux Frames for Codimension 2 Submanifolds contained in Hypersurfaces
考虑余维$1$子流形$N^n\subset M^{n+1}$，其中$M^{n+1}\subset\mathbb{R}^{n+2}$是超曲面。$M$的切空间沿$N$的包络推广了曲面沿曲线的切可展曲面的概念。在本文中，我们研究这些包络的奇异性。有一些重要的子流形的例子，允许一个向量场切于$M$和横截于$N$，其在$N$的任何方向上的导数包含在$N$中。在这种情况下，我们可以在$N$上构造横截平面丛和仿射度量，使其具有等仿射和非极性的理想性质。此外，这个横截丛与经典的Transon平面概念相一致。但我们也给出了一个明确的例子，一个子流形，不允许一个向量场具有上述性质。
In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the rel…
Let $\mathfrak{M}(Σ)$ be an open and connected subset of the space of hyperbolic metrics on a closed orientable surface, and $\mathfrak{M}(M)$ an open and connected subset of the space of metrics on an orientable manifold of dimension at least $3$. We impose conditions on $M$ and $\mathfrak{M}(M)$, which are often satisfied when the metrics in $\mathfrak{M}(M)$ have non-positive curvature. Under these conditions, the data of a homotopy class of maps from $Σ$ to $M$ gives $\mathfrak{M}(Σ)\times \mathfrak{M}(M)$ the structure of a space of harmonic maps. Using transversality theory for Banach ma…
On a compact foliated Riemannian manifold with some transversal curvature conditions, there are no nontrivial basic harmonic forms (M. Min-Oo et al., J. Reine Angew. Math. 415 (1991). In this paper, we extend the above facts to a complete foliated Riemannian manifold.
In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the shape operator and the normal Jacobi operator of the hypersurface commute.
In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.
Monotone quantities involving a weighted $σ_k$ integral along inverse curvature flows
给出了欧几里得空间中逆曲率流的一族沿光滑解的单调量。对于具有正k次中曲率的闭超曲面，我们也得到了一个相关的几何不等式。
In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic quantities under the mean curvature flow.
Bounding sectional curvature along a Kähler-Ricci flow
若正第一Chern类紧Kähler $n$ -流形$n\geq 3$上的正规化Kähler-里奇流$g(t),t\in[0,\infty),$满足$g(t)\in 2πc_{1}(M)$且有$L^{n}$曲率算子一致有界，则曲率算子沿流也一致有界.因此，流将沿着一个子序列收敛到Kähler-Ricci孤子。
We construct weak solutions for the evolution of hypersurfaces along their inverse space-time mean curvature in asymptotically flat maximal initial data sets. As the speed of the new flow is given by a space-time invariant, it can detect both future- and past-trapped apparent horizons. The weak solution extends concepts developed by Huisken-Ilmanen for inverse mean curvature flow and by Moore for inverse null mean curvature flow.
Constant mean curvature hypersurfaces condensing along a submanifold
给定余维数大于1的任意非退化k维极小子流形K，我们证明了常中曲率子流形族的存在性，这些子流形族的中曲率从族中的一个成员到另一个成员是不同的，它们“凝聚”到K.特别地，我们的结果证明了任何黎曼流形中具有非平凡拓扑的常中曲率超曲面的存在性。
A New Monotone Quantity along the Inverse Mean Curvature Flow in $\mathbb R^n$
我们发现了$\mathbb R^n$中逆中曲率流的一个新的沿光滑解的单调递增量。作为应用，我们得到了平均凸的星形超曲面的一个尖锐的几何不等式，它将超曲面所包围的体积与超曲面的加权总中曲率联系起来。
We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.
In this article, we consider compact surfaces $Σ$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $Γ=\partialΣ\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the warped product $ \mathbb{M}\times_f\mathbb{R} $, here $ \mathbb{M} $ denotes a Hadamard surface. We obtain height estimate for a such surface $Σ$ having positive constant mean curvature involving the area of a part of $Σ$ above of $ \mathbb{M} _0$ and the volume it bounds. Also we give general conditions for the existence of rotationally-invariant topological spheres havin…
A scalar curvature bound along the conical Kähler-Ricci flow
从一个模型圆锥Kähler度量出发，在扭曲典范丛上假设半满条件，证明了圆锥Kähler-里奇流解的一致数量曲率界.在证明中，我们还建立了势及其时间导数的一致估计。
Singular metrics with nonnegative scalar curvature and RCD
本文证明了在特定流形上具有孤立奇点的均匀Riemannian度量若具非负数量曲率则Ricci平坦且可延拓，从而证实了Schoen猜想。
The mean curvature flow along the Kähler-Ricci flow
设$(M,\overline{g})$为Kähler曲面，$Σ$为$M$中的浸入曲面.在$M$中$Σ$的Kähler角由Chern-Wolfson \cite{CW}引入。令$(M,\overline{g}(t))$沿里奇流演化，$(M,\overline{g}(t))$中的$Σ_t$沿中曲率流演化.我们证明了Kähler角$α(t)$满足演化方程：$$ (\frac{\partial}{\partial t}-Δ)\cosα=|\overline\nabla J_{Σ_t}|^2\cosα+R\sin^2α\cosα, $$其中$R$是$(M, \overline{g}(t))$的标量曲率。该方程意味着，如果初始曲面是辛的（拉格朗日），那么沿着流动，$Σ_t$在每个时刻$t$总是辛的（拉格朗日），我们称之为辛（拉格朗日）Kähler-Ricci中曲率流。本文主要研究辛Kähler-Ricci中曲率流。
An area growth argument for null mean curvature flow along the standard de Sitter lightcone
该研究探讨德西特标准光锥上的零平均曲率流，通过尺度变换将其与亏格0曲面的2d-Ricci流关联，进而分析流的奇点形成、渐近行为与古老解。
Harmonic Mean Curvature Lines on Surfaces Immersed in R3
考虑定向表面沉浸在$\mathbb R^3.$与他们有关，这里研究了对横向叶理与奇点，定义在椭圆区域，其中高斯曲率$\mathcal K$，由产品的主曲率$k_1, k_2$是积极的。叶理的叶片是调和平均曲率的线，也称为特征线或对角线，沿着该线，浸入的法曲率由${\mathcal K}/{\mathcal H}$给出，其中$ {\mathcal H}=({k_1}+k_2)/2$是算术平均曲率。也就是说，${\mathcal K}/{\mathcal H}=((1/{k_1} + 1/{k_2})/2)^{-1}$是浸没的主曲率$k_1, k_2$的调和平均。叶理的奇点是脐点和抛物线，其中$k_1 = k_2$和${\mathcal K} = 0$分别是。本文确定了叶理脐点附近的调和平均曲率线、抛物曲线和调和平均曲率圈的结构稳定模式。建立了这些模式的通用性。这提供了三个基本的本地成分，以建立充分的条件，可能也是必要的，为调和平均曲率结构稳定性的浸没表面。这项研究，概述了对年底的文件，是一个自然的模拟和补充，以前进行的作者的算术平均曲率和渐近结构稳定的浸没表面。
Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $β\in (0,1)$. We show that in any positive co-homology class on $X$ there is a Kähler metric with cone angle $2πβ$ along $D$ which has bounded Ricci curvature. We use this result together with the Aubin-Yau continuity method to give an alternative proof of a well-known existence theorem for Kahler-Einstein metrics with cone singularities.
Evolution equations of curvature tensors along the hyperbolic geometric flow
本文研究了由Kong和Liu [KL]引入的双曲几何流$\frac{\partial^2}{\partial t^2}g(t)=-2Ric_{g(t)}$.当黎曼度量演化时，它的曲率也会演化。利用S.Brendle [Br，BS]的方法和思想，我们导出了Levi-Civita联络的发展方程和沿双曲几何流的曲率张量.与文献[DKL 1]中的局部法坐标方法不同，本文的方法和结果是以整体张量形式计算和表示的。此外，我们还证明了在有限时间内产生奇点的双曲几何流的任何解都具有无界的里奇曲率。
In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in distributional sense provided that the initial metric is $W^{1,p}$ for some $n<p\le \infty$. As an application, we use this result to study the relation between Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in di…
A dynamical interpretation of the profile curve of cmc Twizzlers surfaces
众所周知，对于任何非零的$M in (-1/4,infty)$，如果我们在平面中的一条直线上滚动二次曲线{（x，y）：4 x^2-y^2/M}=1}，然后我们围绕这条直线旋转焦点的轨迹，那么我们获得具有中曲率1的旋转曲面D（M）。当M<=0时，D（M）是嵌入的，称之为波状体;当M>0时，D（M）不是嵌入的，称之为结节体。这些表面被称为Delaunay，它们由圆组成。上述结构中焦点的轨迹称为Delaunay轮廓曲线，它横切于圆。另一个众所周知的家庭恒定的中曲率表面是Twizzlers，他们是叶状的螺旋线，我们可以自然地定义一个轮廓曲线是横向的螺旋线。为了使本摘要中的演示更容易，我们将只考虑中曲率为1的Twizzlers。在本文中，我们将证明，如果我们在平面中的一条线上滚动Twizzler的轮廓曲线，并且同时，我们以相同的速度移动该线中的点，使得轮廓曲线的滚动运动看起来像是放置在跑步机上，那么，由轮廓曲线的原点形成的轨迹是闭合心形曲线HS（M，w）={（x，y）：x^2+y^2+y/（1+ w^2x^2）=M}，其中M>-1/4且w>0。利用这种动力学解释，我们将证明Twizzlers的几个性质
Collapsing and Dirac-Type Operators
本文研究了一个几何Dirac型算子在有界直径和有界截面曲率的坍缩下的谱极限。在光滑极限空间B的情形下，我们证明了谱的极限是由B上某个一阶微分算子的谱给出的，它可以用超连通构造.在一般极限空间X的情况下，我们将极限算子表示为G-空间Y上的横截椭圆算子，其中X = Y/G。作为应用，我们给出了Dirac型算子平方的第k个特征值在直径和截面曲率方面不具有一致上界的流形的一个特征.我们还给出了有限体积流形上具有负截面曲率的Dirac型算子的本质谱公式。
Random k-Surfaces
主要结果是构造了紧致双曲3-流形的所有k-曲面空间上的满支撑遍历横截测度。这个空间是一个层合空间，它的每一片叶子都被标识为一个“完整”的k-曲面，即一个常数（外在）曲率k的曲面，其中k属于] 0，1 [。在其他地方，这个空间已被证明具有混沌性质模仿那些的测地流。这种构造是通过对空间进行编码来实现的。
Parallel mean curvature surfaces with constant contact angle along free boundaries
本文研究了具有常接触角的平行平均曲率曲面，通过分类和构造例子验证了相关定理的准确性。
Generic neck pinch singularities along 2D Lagrangian mean curvature flow
本文研究了二维拉格朗日均曲率流中的非退化颈部 pinch 神经奇点，证明其稳定性并回答了相关问题，同时引入了无法发生的非退化泪滴状奇点。
Nondegenerate neck pinches along the mean curvature flow
本文证明对于一般光滑紧致初始曲面，三维欧氏空间中的平均曲率流在首次奇异时刻会出现球形或非退化颈缩奇点，且这些奇点在时空中孤立，并据此给出了一种构造带手术的平均曲率流的新方法。
Let $L_t$ be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi-Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. Furthermore, if $L_0$ is a sphere that is stable in the sense of Thomas-Yau, then such a singularity cannot form.
Singularity formation along the line bundle mean curvature flow
本文研究线丛平均曲率流，构造了有限时间奇点以否定一般长时间存在性，并在超临界相位与Calabi对称性下证明长时间存在性，进而给出半稳定情形中无限时间奇点的例子。
We give a new proof of the uniformization theorem of the leaves of a lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversaly continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.
In this note, we establish a local maximum principle along Ricci flow under scaling invariant curvature condition. This unifies the known preservation of nonnegativity results along Ricci flow with unbounded curvature. By combining with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard's localized version of a maximum principle given by R. Bamler, E. Cabezas-Rivas, and B. Wilking on the lower bound of curvature conditions.
In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.
In this paper, we discuss centroaffine geometry of polygons in $3$-space. For a polygon $X$ that is locally convex with respect to an origin together with a transversal vector field $U$, we define the centroaffine dual pair $(Y,V)$ similarly to [6]. We prove that vertices of $(X,U)$ correspond to flattening points for $(Y,V)$ and also that constant curvature polygons are dual to planar polygons. As an application, we give a new proof of a known $4$ flattening points theorem for spatial polygons.
The relations among two transversal submanifolds and global manifold
在黎曼几何中，讨论了两个横截子流形与整体流形之间的关系。通过用一个子流形的切向量代替另一个子流形的法向量，将这三个流形的度量张量、Christoffel符号和曲率张量联系在一起。当两个切向量的内积为零时，这些关系的某些推论给出了传统子流形理论中最重要的第二基本形式和Gauss-Codazzi方程。作为特殊情况，考虑了整体流形为欧氏流形的情形。指出要得到子流形中物质场的非零能量动量张量，必须有上述内积和另一子流形的贡献。一般来说，一个子流形通过上述内积与另一个子流形的物质场密切相关。这一结论与Kaluza-Klein理论一致，并可将弦理论和D-膜理论中的流形直积模型推广到更一般的情形--非直积流形。
We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in dimension $3$ a general contruction of Type I neckpinching and clarify the relationship between stable spheres and non-trivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of sing…
Geometric Mean Curvature Lines on Surfaces Immersed in R3
这里研究了对横向叶理与奇点，定义在椭圆区域（其中高斯曲率$\mathcal K$是积极的）的定向表面沉浸在$\mathbb R^3$。叶理的叶是几何平均曲率的线，沿着该线的法曲率由$\sqrt {\mathcal K}$给出，其是浸入的主曲率$ k_1, k_2$的几何平均曲率。叶理的奇点是脐点和抛物线}，其中$ k_1 = k_2$和${\mathcal K} = 0$分别。确定了叶理脐点附近的几何平均曲率线、抛物线和几何平均曲率圈的结构稳定模式。建立了这些模式的通用性。这为建立几何平均曲率结构稳定性的充分条件提供了三个必要的局部成分，这些条件可能也是必要的。这项研究，概述了在文件的最后，是一个自然的模拟和补充的算术平均曲率和渐近结构稳定性的浸没表面的作者以前研究。
Half-space theorems for translating solitons of the r-mean curvature flow
本文研究了r-均曲率流中translating soliton的半空间定理，证明了在特定生长条件下，此类soliton无法完全包含在某些半空间或锥体中。
In this paper we prove a local curvature estimate for the $κ$-LYZ flow over Kähler manifolds introduced in [FGP1] and [LYZ]. In particular, we generalize the long time existence of the flow.
Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds
给定$\mathbb{R}^{m+1}$的一个整环$Ω$和$\pa Ω$的一个$k$维非退化极小子流形$K$，证明了存在一族嵌入的常中曲率超曲面，它们的中曲率趋于无穷大，并沿$K$集中，与$\partial Ω$垂直相交.
A Minimal Lamination of the Unit Ball with Singularities along a Line Segment
我们在欧氏空间的单位球中构造了一列紧致嵌入极小圆盘，它们的边界在球的边界上，并且曲率在从原点延伸的垂直轴的线段上的每一点处爆炸.进一步研究了最小极限层合的横向结构，发现了沿线段的可去奇点和原点的不可去可去奇点。这扩展了结果的冷和Minicozzi，他们建造了一个序列的曲率吹只在中心的球，院长的建设一个序列的曲率吹在一个规定的离散集的点，以及经典的情况下序列的重新缩放螺旋面的曲率吹沿整个垂直轴。
Linear transports along paths in vector bundles V. Properties of curvature and torsion
本文给出了线性输运沿路径的曲率和挠率的几何解释。一些（Bianchi型）的身份满足这些数量的推导。所得到的结果包含作为特殊情况的相应的经典的曲率和挠的线性连接。
In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of \emph{Bismut-Griffiths-positivity} for complex Hermitian non-Kähler manifolds. Since the Kähler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these exampl…
Lines of Mean Curvature on Surfaces Immersed in R3
与浸入R^3中的定向曲面相关联，这里研究了定义在椭圆区域上的具有奇点的横向叶理对，其中高斯曲率K由浸入的主曲率k_1、k_2的乘积给出，是正的。叶理的叶是M-平均曲率的线，沿此线浸入的法曲率由[k_1，k_2]中的函数M = M（k_1，k_2）给出，称为M-中曲率，其性质扩展并统一了算术H=（k_1+k_2）/2的性质，几何K^{1/2}和调和K/H=（（1/{k_1} + 1/{k_2}）/2）^{-1}经典中曲率。叶理的奇点分别是脐点和抛物线，其中k_1 = k_2，K = 0。本文确定了脐点附近的M-中曲率线、抛物线和M-中曲率圈（叶理的周期叶）的模式，它们在浸入的小扰动下是结构稳定的。这些模式的通用性也建立。这些模式提供了三个基本的局部成分，以建立浸没表面的M-中曲率结构稳定性的充分条件，这可能也是必要的。这构成了一个自然的统一和补充的结果，作者以前获得的算术，渐近，几何和谐波，经典的情况下，中曲率结构稳定性。
We investigate the scalar curvature behavior along the normalized conical Kähler-Ricci flow $ω_t$, which is the conic version of the normalized Kähler-Ricci flow, with finite maximal existence time $T<\infty $. We prove that the scalar curvature of $ω_t$ is bounded from above by $C/(T-t)^2$ under the existence of a contraction associated to the limiting cohomology class $[ω_T]$. This generalizes Z. Zhang's work to the conic case.
In this paper, we consider the stability of quermassintegral inequalities along a inverse curvature flow. We choose a special rescaling of the flow such that the $k$-th quermassintegral is decreasing and the $k-1$-th quermassintegral is preserved. Along this rescaled flow, we prove that the decreasing rate of the $k$-th quermassintegral is faster than the Fraenkel asymmetry of the domain when approaching to the sphere. This leads to the stability inequality of quermassintegral inequalities for nearly spherical sets using the flow method.
Normal curvature bounds along the mean curvature flow
设$(M^n,g_0)$和$(\bar{M}^{n+1},\bar{g})$是完备黎曼流形，其中$k \le 2$具有$|\bar{\nabla}^k\bar{Rm}|\le \bar{C}$，且存在具有有界第二基本形式的等距浸入$F_0: M^n \rightarrow \bar{M}^{n+1}$.设$F_t: M^n \rightarrow \bar{M}^{n+1}$（$t\in [0,T]$）是一族由具有初始数据$F_0$和一致有界第二基本形式的中曲率流演化的浸入.我们证明了浸入$F_t$的法曲率的上确界和下确界以有界的速率变化。这是一个类似的结果荣和Kapovitch对里奇流。
Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow
本文研究带孤立锥形奇点的空间上的Ricci de Turck流，证明其保持Ricci曲率初始正则性与附加假设下的数量曲率正性，为奇异Ricci流的曲率算子正性研究奠定基础。
We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the $\infty$-section exist if and only if the log Futaki invariant vanishes on the fibrewise $\mathbb{C}^*$-action, giving a supporting evidence t…
Higher-order Ricci estimates along immortal Kähler-Ricci flows
研究Kähler-Ricci流在中间Kodaira维数紧致Kähler流形上的高阶曲率估计，证明Ricci曲率在C^1下一致有界，Ricci曲率拉普拉斯在C^0下有界，标量曲率在C^2下有界，识别高阶曲率界几何障碍，非零导致Ricci曲率三阶导数以e^{t/2}速率发散。
We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. the $ν$-th order Fourier coefficients of eigenfunctions $e_λ$ over a closed smooth curve $γ$ which satisfies a natural curvature condition, go to 0 at the rate of $O((\logλ)^{-1/2})$, if $0<|ν|/λ<1-δ$, for any fixed $0<δ<1$. Our result implies, for instance, the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of $O((\logλ)^{-1/2})$. A direct corollary of our results and the QER theorem of Toth and Zelditch is that for a geodesi…
In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi-Maupertuis metric. This characterization works for the $N$-body problem with general masses and any $1/r^α$ potential with $α> 0$. We also observe dynamical consequences of these curvature values for relative equilibrium solutions. These curvature methods work well for strong forces ($α\ge 2$).
An Interpolating Curvature Condition Preserved By Ricci Flow
本文首先建立了里奇流上的非负曲率不变量和2-非负曲率不变量之间的插值曲率不变量。然后，沿着里奇流导出了$(λ_1, λ_2)$ -非负性的一个相关的强最大值原理.在此基础上，我们得到了具有$(λ_1,λ_2)$ -非负曲率算子的流形的一个刚性性质。
Linear transports along paths in vector bundles. III. Curvature and torsion
定义了分别沿向量丛和微分流形切丛中路径的线性输运的曲率和挠率，并推导了它们的某些性质。
This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on mean curvature flow shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.
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