This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green's function is proved as a tool.
We show that the isoperimetric profile $h_{g(t)}(ξ)$ of a compact Riemannian manifold $(M,g)$ is jointly continuous when metrics $g(t)$ vary continuously. We also show that, when $M$ is a compact surface and $g(t)$ evolves under normalized Ricci flow, $h^2_{g(t)}(ξ)$ is uniform Lipschitz continuous and hence $h_{g(t)}(ξ)$ is uniform locally Lipschitz continuous.
We study the problem of convergence of the normalized Ricci flow evolving on a compact manifold $Ω$ without boundary. In \cite{KS10, KS15} we derived, via PDE techniques, global-in-time existence of the classical solution and pre-compactness of the orbit. In this work we show its convergence to steady-states, using a gradient inequality of Łojasiewicz type. We have thus an alternative proof of \cite{ha}, but for general manifold $Ω$ and not only for unit sphere. As a byproduct of that approach we also derive the rate of convergence according to this steady-sate being either degenerate or non-d…
The main purpose of this paper is to show that a normalized non-steady gradient Ricci soliton (M,g,f,λ) of dimension n is trivial if and only if its scalar curvature S satisfies the equality S=λ(f+(n/2)).
Consider a Riemannian manifold $(M^{m}, g)$ whose volume is the same as the standard sphere $(S^{m}, g_{round})$. If $p>\frac{m}{2}$ and $\int_{M} \left\{ Rc-(m-1)g\right\}_{-}^{p} dv$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^{m}, g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.
Eigenvalue monotonicity for the Ricci-Hamilton flow, revised version
这是我们的短文[arxiv.math.DG/0403065]的一个修正版本，其中我们讨论了紧致或完备非紧致黎曼流形上Laplacian算子的特征值对Ricci-Hamilton流的单调性。我们证明了当数量曲率有一个非负的下界且Einstein张量不太负时，紧区域上Lapacian算子的特征值是非减的.这一结果对研究Ricci-Hamilton流的爆破模型是有用的。我们注意到，利用我们的思想，D.Kokotov和D.Korotkin [Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian，Letters in数学物理，71（3）（2005）241-242]可以给出一个简单的证明，证明任意亏格g的紧致Riemann曲面上光滑度量中Laplace算子的行列式在归一化里奇流下单调增长.结合Hamilton和B.Chow的结果，在正规化里奇流的作用下，光滑度量渐近地趋向于常曲率度量，这导致Osgood-Phillips-Sarnak定理的简单证明，该定理指出在具有固定共形类和固定体积的光滑度量类内，拉普拉斯算子的行列式在常曲率度量上是最大的。人们可以考虑其他几何流和其他椭圆算子的类似问题，如Yamabe流或Kahler-里奇流。
In this paper, authors have established Chen's inequalities for the submanifolds of quaternionic Kaehler manifolds characterized by Ricci quarter-symmetric metric connection. Other than these inequalities, we have also derived generalized normalized Casorati curvature inequalities for the same submanifolds.
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.
We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds $V_2\mathbb{R}^n$ and $V_{1+k_2}\mathbb{R}^n$, with $n = 1+k_2+k_3$. We use techniques from the theory of differential equations, in particular the Poincaré compactification.
Beijing Yanqi Lake Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）；Fudan University（复旦大学）；Tsinghua University（清华大学）；Renmin University of China（中国人民大学）
We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up to the time where the norm of the Riemann tensor diverges. Restricting to initial metrics which belong to this class and are rotationally symmetric, we prove that if the sectional curvature in planes tangent to the orbits of symmetry is initially nonpositive, the flow starting from such an initial metric exists for all time. Moreover, if the sectional curva…
We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian \cite{ST3} in case of projective varieties with klt singularities. We also prove that the normalized K"ahler-Ricci flow will converge to the \ka-Einstein metric with negative Ricci curvature on semi-log canonical models in the sense of currents. Finally we also construct K"ahler-Ricci flow solutions performing divisorial contractions and flips with log canonical singularities.
It is well known that the Kähler-Ricci flow on a Kähler manifold $X$ admits a long-time solution if and only if $X$ is a minimal model, i.e., the canonical line bundle $K_X$ is nef. The abundance conjecture in algebraic geometry predicts that $K_X$ must be semi-ample when $X$ is a projective minimal model. We prove that if $K_X$ is semi-ample, then the diameter is uniformly bounded for long-time solutions of the normalized Kähler-Ricci flow. Our diameter estimate combined with the scalar curvature estimate in [34] for long-time solutions of the Kähler-Ricci flow are natural extensions of Perel…
