Non Parallelism of Ricci tensor of Pseudo-cylindric Metrics

Cylindricity of complete Euclidean submanifolds with relative nullity

We show that a complete Euclidean submanifold with minimal index of relative nullity $ν_0>0$ and Ricci curvature with a certain controlled decay must be a $ν_0$-cylinder. This is an extension of the classical Hartman cylindricity theorem.

Rotational symmetry of Ricci solitons in higher dimensions

Let (M,g) be a steady gradient Ricci soliton of dimension n \geq 4 which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that (M,g) is rotationally symmetric. In particular, our result applies to steady gradient Ricci solitons in dimension 4 which are κ-noncollapsed and have positive isotropic curvature.

On rotationally invariant shrinking gradient Ricci solitons

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are, respectively, the round, flat, and standard cylindrical metrics.

On cylindrically bounded $H$-Hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R}$

We show that $H$-hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R}$ contained in a vertical cylinder and with Ricci curvature with strong quadratic decay have mean curvature $| H| > (n-1)/n$.

Ricci flow on quasiprojective manifolds II

We study the Ricci flow on complete Kaehler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial infinity, are transversely hyperbolic. In the present paper we consider three different types of spatial asymptotics: cylindrical, bulging and conical. We show that in each case, the asymptotics are preserved by the Kaehler-Ricci flow. We address long-time existence, parabolic blowdown limits and the role of the Kaehler-Ricci flow on the divisor.

具有循环平行里奇张量的空间的直接证明

Direct proofs for spaces with cyclic parallel Ricci tensor

该研究为梯度里奇孤子、带边界紧致拟爱因斯坦流形及具循环平行里奇张量的临界空间的刚性结果提供直接证明，通过推导张量恒等式等方法得到相关刚性结论与里奇平行性结果。

轨道流形上正曲率稳态梯度Ricci孤立子的刚性

Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

本文研究轨道流形上正曲率稳态梯度Ricci孤立子的刚性，证明其数量曲率非负，并在正曲率算子或正截面曲率条件下，此类孤立子必须是Bryant孤立子的有限商。

Ricci 流颈部收缩的指数渐近

Exponential Asymptotics of Ricci Flow Neckpinch

本文研究紧致流形上 Ricci 流的柱形颈部收缩奇点，在 Type-I 假设下证明先前构造的渐近轮廓是唯一的可能。

正$τ$-双Ricci曲率与双曲空间中带手术的平均曲率流

Positive $τ$-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space

该研究证明双曲空间中满足正$τ$-双Ricci曲率的光滑闭连通浸入超曲面存在带有限次手术且可终止的平均曲率流，并推导了相应流形的微分同胚类型与紧区域的拓扑性质。

具有$\mathcal{L}^{1}$衰减曲率的稳态溶iton

Steady soliton with $\mathcal{L}^{1}$ decay curvature

本文基于梯度黎曼度量溶iton的标量曲率界和调和坐标下界，建立紧致性定理，并证明非塌陷极限的正则部分光滑，以及稳态梯度黎曼度量溶iton在Ricci曲率$\mathcal{L}^{1}$衰减下的渐近柱状性。

线性体积增长的Ricci平坦流形渐近极限的唯一性

Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth

本文在自然曲率与横截面假设下，证明了线性体积增长Ricci平坦流形渐近极限的唯一性与指数收敛，并开发常平均曲率叶状结构处理圆柱形新出现的零模与负模，加强并回答了相关猜想。

Kähler Finsler manifolds with curvatures bounded from below

We obtain a partial parallelism of the complex structure on Kähler Finsler manifolds. As applications, we prove Synge-Tsukamoto theorem and Bonnet-Myers theorem for positively curved Kähler Finsler manifolds. Moreover, we generalize a comparison theorem due to Ni-Zheng by introducing the notion of orthogonal Ricci curvature to Kähler Finsler geometry.

高维稳态和扩展溶iton的曲率估计

Curvature estimates for steady and expanding solitons in higher dimensions

本文研究了高维稳态和扩展溶iton的曲率估计，改进了之前的结果并去除了非负里奇曲率假设，同时证明了相关几何结构的存在性和唯一性。

Curvature properties of Melvin magnetic metric

This paper aims to investigate the curvature restricted geometric properties admitted by Melvin magnetic spacetime metric, a warped product metric with $1$-dimensional fibre. For this, we have considered a Melvin type static, cylindrically symmetric spacetime metric in Weyl form and it is found that such metric, in general, is generalized Roter type, $Ein(3)$ and has pseudosymmetric Weyl conformal tensor satisfying the pseudosymmetric type condition $R\cdot R-Q(S,R)=\mathcal L' Q(g,C)$. The condition for which it satisfies the Roter type condition has been obtained. It is interesting to note t…

Linear Connections and Curvature Tensors in the Geometry of Parallelizable Manifolds

In this paper we discuss curvature tensors in the context of Absolute Parallelism geometry. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection. Using the Bianchi identities some other identities are derived from the expressions obtained. These identities, in turn, are used to reveal some of the properties satisfied by an intriguing fourth order tensor which we refer to as Wanas tensor. A further condition on the canonical connection is imposed, assuming it is semi-symmetric. The formulae thus obtained, together with other form…

The Steklov spectrum and coarse discretizations of manifolds with boundary

We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a manifold M of this class, we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to M, with constants depending only on the dimension and bounds on curvature and injectivity radius. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of the manifold M and those of its di…

乘积空间中的双守恒与双调和超曲面

Biconservative and Biharmonic Hypersurfaces in Product Spaces

本文研究乘积空间中的双守恒与双调和超曲面，给出等参与常角情形的完整描述，并证明分类、刚性及全局不存在性定理。

Complete Calabi-Yau metrics from P^2 # 9 \bar P^2

Let $X$ denote the complex projective plane, blown up at the nine base points of a pencil of cubics, and let $D$ be any fiber of the resulting elliptic fibration on $X$. Using ansatz metrics inspired by work of Gross-Wilson and a PDE method due to Tian-Yau, we prove that $X \setminus D$ admits complete Ricci-flat Kähler metrics in most de Rham cohomology classes. If $D$ is smooth, the metrics converge to split flat cylinders $\R^+ \times S^1 \times D$ at an exponential rate. In this case, we also obtain a partial uniqueness result and a local description of the Einstein moduli space, which con…