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by
Jia-yong Wu, Peng Wu, William Wylie
, 2014

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...|2 R2 = 1 24 . If (M, g, f) is a gradient shrinking Ricci soliton, it follows directly from Theorem 1.1. If (M, g, f) is a gradient steady Ricci soliton, then R ≡const implies that R ≡ 0 (see [21] or =-=[44]-=-), and by Proposition 4.3 in [39], (M, g, f) is a finite quotient of M × C, where M is a flat Riemann surface. If (M, g, f) is a gradient expanding Ricci soliton, then by Proposition 3.2 and Lemma 3.1...

by
M. Buzano, A. S. Dancer, M. Gallaugher, M. Wang

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...w describe some general results about complete cohomogeneity one steady solitons of gradient type. Some of these results can be deduced from theorems about general solitons found in e.g., [FR], [MS], =-=[Wu]-=-, [WeiWu]. However, in the cohomogeneity one situation, the statements sometimes take on a stronger or more precise form, which will be useful for checking asymptotic behaviour in numerical studies. W...

by
Guofang Wei, Peng Wu

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.... Though one can make product of two shrinking ones, but all trivial shrinking ones are compact. So it will not give a constant direction by taking a product. Munteanu-Sesum [8] and the second author =-=[13]-=- independently showed that the infimum of f does decay linearly. In fact −r ≤ inf y∈∂B(x,r) f(y)− f(x) ≤ −r + √ 2n( √ r + 1), r ≫ 1. (1.4) In particular, lim infy→∞R(y) = 0, see also [6, 1]. We note t...

by
Huai-dong Cao, Xin Cui

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...oved in [9], as well as a key scalar curvature lower bound R ≥ c/f shown in [15] are crucial in their work. While gradient steady Ricci solitons in general don’t share these two special features (cf. =-=[31, 24, 30]-=- and [15, 20]), some of the arguments in [25] can still be adapted to prove certain curvature estimates for two classes of gradient steady solitons. Our main results are: Theorem 1.1. Let (M4, gij , f...

by
Jingyi Chen, Yue Wang

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... in the proof of Proposition 2.1 is no longer valid for steady gradient Ricci solitons due to different behaviour of f (typically f tends to −∞ along a sequence of points xk that go to infinity [17], =-=[26]-=-). Alternatively, a powerful way to prove Liouville type theorems for positive harmonic functions on complete manifolds with non-negative Ricci curvature is via Yau’s gradient estimate [24]. The p-har...

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