Geometry of Complete Gradient Shrinking Ricci Solitons
نویسنده
چکیده
The notion of Ricci solitons was introduced by Hamilton [24] in mid 1980s. They are natural generalizations of Einstein metrics. Ricci solitons also correspond to self-similar solutions of Hamilton’s Ricci flow [22], and often arise as limits of dilations of singularities in the Ricci flow. In this paper, we will focus our attention on complete gradient shrinking Ricci solitons and survey some of the recent progress, including the classifications in dimension three, and asymptotic behavior of potential functions as well as volume growths of geodesic balls in higher dimensions.
منابع مشابه
The volume growth of complete gradient shrinking Ricci solitons
We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature. A complete Riemannian manifold M of dimension n is called gradient shrinking Ricci soliton if there exists f ∈ C (M) and a constant ρ > 0 such that Rij +∇i∇jf = ρgij , where Rij denotes the Ricci...
متن کاملGeometry of compact shrinking Ricci solitons
Einstein manifolds are trivial examples of gradient Ricci solitons with constant potential function and thus they are called trivial Ricci solitons. In this paper, we prove two characterizations of compact shrinking trivial Ricci solitons. M.S.C. 2010: 53C25.
متن کاملSharp Logarithmic Sobolev Inequalities on Gradient Solitons and Applications
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 invari...
متن کاملA Characterization of Noncompact Koiso-type Solitons
We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton’s equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometr...
متن کاملFour-dimensional Gradient Shrinking Solitons with Positive Isotropic Curvature
We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S4 or a quotient of S3 × R. This gives a clean classification result removing the earlier additional assumptions in [14] by Wallach and the second author. The proof also gives a classification result on gradient shrinking Ricci solitons with nonnegative isotropic c...
متن کامل