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Register a new userRigidity of the round cylinders in Ricci shrinkers
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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 $S^{n-1}times mathbb R$.
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Let $(M, g, f)$ be a $4$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+ abla^2f=lambda g$, where $lambda$ is a positive real number. We prove that if $M$ has constant scalar curvature $S=2lambda$, it must be a quotient of $mathbb{S}^2times mathbb{R}^2$. Together with the known results, this implies that a $4$-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton $Bbb{R}^4$, $Bbb{S}^{2}timesBbb{R}^{2}$ or $Bbb{S}^{3}timesBbb{R}$.
We classify four-dimensional shrinking Ricci solitons satisfying $Sec geq frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $mathbb{R}^{4}$ (and quotients), $mathbb{S}^{4}$, $mathbb{RP}^{4}$ or $mathbb{CP}^{2}$ with their standard metrics.
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Chengs maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{3}$ with almost maximal width must be nearly round.
Let $M$ be a compact $n$-manifold of $operatorname{Ric}_Mge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $rho>0$ such that for any $xin M$, the open $rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_rho,x^*)to (B_rho(x),x)$, has the maximal volume i.e., the volume of a $rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].
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