Maximum principles and gradient Ricci solitons

Fernández-López, Manuel; Garcı́a-Rı́o, Eduardo

doi:10.1016/j.jde.2011.03.020

Cited by 23 publications

(16 citation statements)

References 11 publications

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“…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.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Wylie

2018

Calc. Var.

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We prove that a four-dimensional gradient shrinking Ricci soliton with δW ± = 0 is either Einstein, or a finite quotient of S 3 × R, S 2 × R 2 or R 4 . We also prove that a four-dimensional cscK gradient Ricci soliton is either Kähler-Einstein, or a finite quotient of M × C, where M is a Riemann surface. The main arguments are curvature decompositions, the Weitzenböck formula for half Weyl curvature, and the maximum principle.

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Section: Proof Of Theorem 11mentioning

confidence: 99%

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Wylie

2018

Calc. Var.

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“…ACKNOWLEDGEMENTS. The authors are grateful to Manuel Fernández-López for having sent them the preprints [6] and [7].…”

Section: Theorem 14 Let (M ∇ F ) Be a Complete Gradient Ricci mentioning

confidence: 99%

Ricci almost solitons

Pigola¹,

Rigoli²,

Rimoldi³

et al. 2012

Annali Scuola Normale Superiore - Classe Di Scienze

124

150

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We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed.

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“…This is a result of Fernández-Lopez and García-Río [8]. By different methods, Munteanu and Sesum [12] and Wu [17] obtained lim inf z→∞ R(z) = 0.…”

Section: Application To Steady Gradient Ricci Solitonsmentioning

confidence: 65%

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

2012

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Following Li and Yau (Acta Math 156:153-201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional J associated to a smooth function φ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.

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Maximum principles and gradient Ricci solitons

Cited by 23 publications

References 11 publications

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Ricci almost solitons

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

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