Recent Progress on Ricci Solitons

By Huaihuo Cao. 2009

374 citations recorded by the source corpus.

Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton's Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons.

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

View Paper: https://www.semanticscholar.org/paper/274748f9db97d710f02d36a8babe1ffebcec44fb

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

On Bach flat warped product Einstein manifolds (2011, methodology): “We refer to the survey paper [Cao 2010] for recent progress on this subject.”
A Compactness Theorem for Complete Ricci Shrinkers (2010, neutral): “Let us start by collecting some basic facts about gradient shrinkers (for a recent survey about Ricci solitons, see [6]).”
Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness (2015, background): “As mentioned earlier, this would show the similarities or differences with the analogous theory for Ricci solitons, which it would be instructive to study (see [9] for a recent survey on Ricci solitons).”
Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces (2012, background): “A survey and further references on the geometry of Ricci solitons may be found in [10].”
Bach-flat gradient steady Ricci solitons (2011, methodology): “For additional examples, we refer the readers to the survey paper [6] by the first author and the references therein.”
Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, methodology): “We refer the reader to the two surveys [13], [14] of Cao for a detailed picture of the current state of the art.”
Smooth metric measure spaces with non-negative curvature (2011, background): “ing Rij + fij = 0.It is known that there exists a positive constant a>0 such that |∇f|2 + S= a2,where Sis the scalar curvature of M.It is also known that S≥ 0 for any gradient steady Ricci soliton [4, 9]. Another useful relation is ∆f+S= 0, which is obtained directly from Ricf = 0 by taking trace”
Geometry of manifolds with densities (2012, methodology): “We shall refer readers to [4] for a recent survey on Ricci solitons.”
Structure of homogeneous Ricci solitons and the Alekseevskii conjecture (2012, methodology): “We refer to [6, 5, 7] and the references therein for further information on Ricci solitons.”