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Register a new userExistence of singular rotationally symmetric gradient Ricci solitons in higher dimensions
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Kin Ming Hui
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By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=frac{da^2}{h(a^2)}+a^2g_{S^n}$ for some function $h$ where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $mathbb{R}^n$ for any $nge 2$. More precisely for any $lambdage 0$ and $c_0>0$, we prove that there exist infinitely many solutions $hin C^2((0,infty);mathbb{R}^+)$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-lambda r-(n-1))$, $h(r)>0$, in $(0,infty)$ satisfying $underset{substack{rto 0}}{lim},r^{sqrt{n}-1}h(r)=c_0$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin.
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It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson-Walker warped product, if the gradient of the potential function is non null, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.
The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless $kappa$-Einstein soliton and is realized on the cotangent bundle of an affine surface.
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 of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton $Bbb{R}^4,$ or $Bbb{S}^{3}timesBbb{R}$, or $Bbb{S}^{2}timesBbb{R}^{2}.$ In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.
The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either $mathds{S}^3$, $erretimesmathds{S}^2$ or $erre^3$, under some very weak conditions on the vector field $X$ generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the $X$-Laplacian on every generic Ricci soliton, without any assumption on $X$.
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 $Bbb{S}^{3}timesBbb{R}$.
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