On the completeness of gradient Ricci solitons

By Zhuhong Zhang. 2008

139 citations recorded by the source corpus.

A gradient Ricci soliton is a triple (M, g, f) satisfying R ij + ∇ i ∇ j f = λg ij for some real number λ. In this paper, we will show that the completeness of the metric g implies that of the vector field ∇f.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cce6f0846c54907abca727a2fdc0801b31e3e423

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Ricci almost solitons (2010, background): “Zhang, [34], where it is shown that the scalar curvature of any gradient Ricci soliton is bounded below.”
Recent Progress on Ricci Solitons (2009, background): “rates. For this reason we often do not distinguish the two. Remark 3.1. If Mn is compact, then V is always complete. But if M is noncompact then V may not be complete in general. Recently Z.-H. Zhang [101] has observed that for any complete gradient (steady, shrinking, or expanding) Ricci soliton gi”
A Compactness Theorem for Complete Ricci Shrinkers (2010, unknown): “…( M n , g, f ), there is a family of Riemannian metrics g ( t ), t ∈ ( −∞ , 1), evolving by Hamilton’s Ricci ﬂow g ( t ) = − 2 Rc g ( t ) with g (0) = g , which is self-similarly shrinking, i.e. g ( t ) = (1 − t ) φ ∗ t g for the family of diﬀeomorphisms φ t generated by − t ∇ f , see [7, 42].”
Remarks on non-compact gradient Ricci solitons (2009, background): “Therefore, by Corollary 13, S− is bounded from above or, equivalently, S∗ = infM S > −∞ (for this conclusion, see also [24]).”
On locally conformally flat gradient steady Ricci solitons (2009, background): “Zhang [22], the gradient vector field V = ∇F is a complete vector field onM .”
Classification results for expanding and shrinking gradient Kähler–Ricci solitons (2019, background): “is picture for an expanding gradient Ricci soliton next. For a complete expanding gradient Ricci soliton (M,g,X) with soliton potential f, the completeness of the vector ﬁeld −X t is ensured by Zhang [Zha09]. We then set g(t) := tϕ∗ tg,t>0, where ϕt is a family of diﬀeomorphisms generated by the ”
Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “g is complete, according to [55]), this construction produces a globally defined ancient solution to the Ricci flow.”
Rotational symmetry of self-similar solutions to the Ricci flow (2012, background): “3 in [28] that (M,g) has positive scalar curvature (see also [5], Proposition 2.”
GRADIENT SHRINKING SOLITONS WITH VANISHING WEYL TENSOR (2008, methodology): “In [Zhang 2008a], the author obtained the following (global) Hamilton–Ivey-type pinching estimate in higher dimension: Theorem.”
Geometry of Complete Gradient Shrinking Ricci Solitons (2009, background): “ﬂat Euclidean space (Rn,δ ij) is a gradient shrinker with potential function f = |x|2/4: ∂ i∂ jf = 1 2 δ ij. (Rn,δ ij,|x|2/4) is called the Gaussian shrinking soliton. 1It was observed by Z.-H. Zhang [43] that V = ∇f is a complete vector ﬁeld if the soliton metric gij is complete 2See alternative pr”
On type-I singularities in Ricci flow (2010, background): “[24]), that ∇f is a complete vector field.”
On the geometry of gradient einstein-type manifolds (2014, unknown): “[25], [28], [39], [22], [38], [48], [37], [40], [10], [7], [14], [42], [11], [13], [8], [6], [33], [5] and references therein).”
Backwards uniqueness of the Ricci flow (2009, background): “, g is a gradient Ricci soliton, then Zhang [Z] has shown that the completeness of the vector field ∇f follows from the completeness of the metric g without the assumption of a curvature bound.”