Classification of complete generic shrinking Ricci solitons with pointwise pinched curvature

Let M n , g , f be a complete gradient shrinking Ricci soliton of dimension n ≥ 3 . In this paper, we study the rigidity of M n , g , f with pointwise pinching curvature and obtain some rigidity results. In particular, we prove that every n -dimensional gradient shrinking Ricci soliton M n , g , f is isometric to ℝ n or a finite quotient of S n under some pointwise pinching curvature condition. The arguments mainly rely on algebraic curvature estimates and several analysis tools on M n , g , f , such as the property of f -parabolic and a Liouville type theorem.

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“…ð39Þ Remark 11. Inequality (39) can also be derived by setting X = ∇f in Lemma 2.5 of [12]; here, we give its proof for the sake of completeness.…”

Section: Advances In Mathematical Physicsmentioning

confidence: 99%

“…In [12], the authors established the following rigidity theorem under pointwise pinching condition ofR ∘ m:…”

Section: Introductionmentioning

confidence: 99%

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Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

Chu

Zhou

2021

Advances in Mathematical Physics

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“…In [CHL17], we have established the elliptic estimate of the norm square of Rm for complete generic Ricci solitons, which will be used directly in this paper as one of the key curvature inequalities. For this purpose, by setting X = ∇f in [CHL17, Lemma 2.5], one can derive the following estimate.…”

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confidence: 99%

Integral pinching characterization of compact shrinking Ricci solitons

2023

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We investigate the pinching problem for shrinking compact Ricci solitons. Firstly, we show that every n-dimensional (n ≥ 4) shrinking compact Ricci soliton (M n , g) is isometric to a finite quotient of S n under an L n/2 -pinching condition. Then we prove that the same result is still true for (M n , g) under an L p -pinching condition for p > 2/n. The arguments rely mainly on algebraic curvature estimates and several important integral inequalities.

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