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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.
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
In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part o
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^{
In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level s
Positively curved Alexandrov spaces of dimension 4 with an isometric circle action are classified up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always