Sharp logarithmic Sobolev inequalities on gradient solitons and applications
By José Antonio Carrillo, Lei Ni. 2008
116 citations recorded by the source corpus.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the {\it asymptotic volume ratio} of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/70da7ce36a4a3ef8b534ea069c11a77de1ea5499
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Geometry of Complete Gradient Shrinking Ricci Solitons (2009, methodology): “concerned with the asymptotic behavior of potential functions, and volume growth rates of geodesic balls. First of all, we shall need the logarithmic Sobolev inequality proved recently by Carrillo-Ni [11] for complete noncompact gradient shrinking solitons. To state their result, instead of normaliz”
- Geometry of manifolds with densities (2012, background): “ +2 Z M ∇f,∇u for any functionR u with compact support in M. Here, we have denoted µ := M e −fdv < ∞. In the case of gradient shrinking Ricci solitons, the inequality can be further simplified, see [8]. Theorem5.3. Let (M,g,f) be a complete smooth metric measure space of dimension n. Assume that R”
- A Compactness Theorem for Complete Ricci Shrinkers (2010, unknown): “…made the wonderful observation that Perelman’s logarithmic Sobolev inequality holds even for noncompact shrinkers without curvature assumptions [11, where the infimum is taken over all ˜ f : M → R ∪ { + ∞} such that ˜ u = e − ˜ f/ 2 is smooth with compact support and M ˜ u 2 dV = (4 π ) n/ 2 .”
- Analysis of weighted Laplacian and applications to Ricci solitons (2011, methodology): “With this normalization of f the Perelman’s invariant is defined by [21, 7]: μ0 := − log (”
- Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, methodology): “, [17], [19], [42]) as indicators of an enforcement of some broader principle of asymptotic rigidity, however, the catalog of nontrivial examples is still exceedingly slim.”
- Expanding solitons with non-negative curvature operator coming out of cones (2010, background): “nown that if an expanding soliton has bounded curvature and Ric > 0, then 0 < AVR < ∞. This was first proved by R.Hamilton for Ric > 0 (see Proposition 9.46 in the book [12]). In the paper [7] this result was generalised to the case that the curvature is not necessarily bounded: see propo”
- On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below (2009, background): “. 1 V(B(o,r)) ≤ C(r +1)n for all r ≥ 0, where B(o,r) is the ball of radius r at center o and V (B(o,r)) denotes the volume of B(o,r) with respect to the metric g. From the proof of Proposition 2.1 in [4], we can obtain the following property; see Lemma 1.1 below. Lemma0.1 Let (Mn,g,f,−1) be a comple”
- On Gradient Ricci Solitons (2009, background): “This is a consequence of the log Sobolev inequality established by Carillo and Ni in [8] and Perelman’s argument in [27].”
- Recent Progress on Ricci Solitons (2009, background): “t; 0 such that the solution is κ-noncollapsed on scales ≤ √ T. In the special case of gradient shrinking solitons, the above result has been improved recently by Naber [77] and further by Carrillo-Ni [19]. Proposition 4.1. (Carrillo-Ni [19]) Let (Mn,g,f) be a gradient shrinking soliton satisfying Eq”