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A characterization of Koiso's typed solitons  [PDF]
Bo Yang
Mathematics , 2008,
Abstract: By extending Koiso's examples to the non-compact case, we construct complete gradient Kahler-Ricci solitons of various types on certain holomorphic line bundles over compact Kahler-Einstein manifolds. Moreover, a uniformization result on steady gradient Kahler-Ricci solitons with non-negative Ricci curvature is obtained under additional assumptions.
A Family of Steady Ricci Solitons and Ricci-flat Metrics  [PDF]
M. Buzano,A. S. Dancer,M. Gallaugher,M. Wang
Mathematics , 2013,
Abstract: We produce new non-K\"ahler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton structures on the vector bundles whose distance sphere bundles are respectively the twistor and ${\rm Sp}(1)$ bundles over quaternionic projective space.
On Potential Function of Gradient Steady Ricci Solitons  [PDF]
Peng Wu
Mathematics , 2011,
Abstract: In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential function must be trivial, and no gradient steady Ricci soliton admits uniformly positive scalar curvature.
Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three  [PDF]
Huai-Dong Cao,Chenxu He
Mathematics , 2014,
Abstract: The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of collapsed positively curved three-dimensional complete gradient steady solitons, with $\mathsf{S}^1$-symmetry, connecting the 3D cigar soliton. In this paper, we make the first initial progress and prove that the infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix A, we show that the 3D cigar soliton is the unique complete nonflat gradient steady Ricci soliton in dimension three that admits two commuting Killing vector fields.
Asymptotic behavior of positively curved steady Ricci Solitons  [PDF]
Yuxing Deng,Xiaohua Zhu
Mathematics , 2015,
Abstract: In this paper, we analyze the asymptotic behavior of $\kappa$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $\kappa$-noncollapsed steady K\"ahler-Ricci soliton with non-negative sectional curvature must be flat.
Bach-flat gradient steady Ricci solitons  [PDF]
Huai-Dong Cao,Giovanni Catino,Qiang Chen,Carlo Mantegazza,Lorenzo Mazzieri
Mathematics , 2011,
Abstract: In this paper we prove that any $n$-dimensional ($n\ge 4$) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in [6] and [9].
On Volume Growth of Gradient Steady Ricci Solitons  [PDF]
Guofang Wei,Peng Wu
Mathematics , 2012,
Abstract: In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.
On Locally Conformally Flat Gradient Steady Ricci Solitons  [PDF]
Huai-Dong Cao,Qiang Chen
Mathematics , 2009,
Abstract: In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons  [PDF]
Huai-Dong Cao,Xin Cui
Mathematics , 2014,
Abstract: In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.
A lower bound for the scalar curvature of certain steady gradient Ricci solitons  [PDF]
Bennett Chow,Peng Lu,Bo Yang
Mathematics , 2011,
Abstract: In this very short note we prove a lower bound for the scalar curvature of certain steady gradient Ricci solitons.
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