On complete gradient shrinking Ricci solitons

By Huaihuo Cao, Detang Zhou. 2009

336 citations recorded by the source corpus.

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/315980e91b1f5ca0fe8e3302af9008f510876679

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Rigidity of gradient Einstein shrinkers (2013, methodology): “We will employ a slight variation of the method exposed in [3].”
On Gradient Ricci Solitons (2009, background): “Our motivation is the results obtained in [2, 4, 8] for shrinking Ricci solitons.”
Remarks on non-compact gradient Ricci solitons (2009, background): “Note that, combining Lemma 2.3 in [ 2 ] with a volume estimate for weighted manifolds, [8,22], it follows that the scalar curvature of a shrinking Ricci soliton is p-integrable, for every p > 0. We are grateful to M. Fernandez-Lopez for pointing out this to us. In the expanding case we shall prove t”
Eigenvalues of the drifted Laplacian on complete metric measure spaces (2013, background): “From [3], we know that for a fixed point p ∈M there exists a constant C such that lim x→+∞ f(x) r2(x) = 1 4 .”
Volume growth, eigenvalue and compactness for self-shrinkers (2011, neutral): “It is worthy to compare the above Theorem with the interesting result of Cao-Zhou on the volume growth of complete gradient shrinking Ricci soliton [3].”
On the geometry of gradient einstein-type manifolds (2014, neutral): “[25], [28], [39], [22], [38], [48], [37], [40], [10], [7], [14], [42], [11], [13], [8], [6], [33], [5] and references therein).”
Gradient Einstein solitons (2012, neutral): “12) In the spirit of [11], we define in the set {r ≥ 0} the function u = 2√f and we set D(s) = {2 < u < s} ⊂ Ω.”
Analysis of weighted Laplacian and applications to Ricci solitons (2011, background): “So by the estimates of f proved in [6, 12] together with (6.”
Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “Here, one might optimistically interpret the sharp estimates now known to hold on the growth of the potential f [17] and the volume of metric balls (see, e.”
Geometry of manifolds with densities (2012, background): “4 recovers the upper bound in [7] and the lower bound in [23] about the volume growth of gradient shrinking solitons.”
Stability and compactness for complete f -minimal surfaces (2012, background): “It was proved by Cao and the third author [1] that there is a positive constant c so that (14) 1 4 (r(x)− c)(2) ≤ f(x) ≤ 1 4 (r(x) + c)(2) for any x ∈ M with r(x) = distM (p, x) ≥ r0, where p is a fixed point in M and c, r0 are positive constants that depend only on m and f(p).”
Bach-flat gradient steady Ricci solitons (2011, background): “…≥ 0 which together imply |∇ ( − f ) | 2 ≤ ( f ). the length argument − The lower bound is an easy consequence of second variation of arc as in, e.g., [8, p.179], applied to the equation Moreover, since |∇ ( − f ) ≤ √− f ≤ r ( x ) + − f ( O ), − f is clearly proper and hence an exhaustion function.”
Geometry of shrinking Ricci solitons (2014, background): “[6] H. D. Cao and D. Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom.”