Recent Progress on Ricci Solitons

Abstract

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.

32 pages; to appear in Proceedings of International Conference on Geometric Analysis (Taipei, July 2007)

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Cited by in corpus (75)

• On Type I Singularities in Ricci flow
• On Bach-flat gradient shrinking Ricci solitons
• Geometry of shrinking Ricci solitons
• A compactness theorem for complete Ricci shrinkers
• Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
• $f$-minimal surface and manifold with positive $m$-Bakry-Émery Ricci curvature
• On the construction of gradient Ricci soliton warped product
• Stability and instability of Ricci solitons
• Ricci solitons in three-dimensional paracontact geometry
• Stability of Einstein metrics under Ricci flow
• Gradient estimates for a nonlinear parabolic equation and Liouville theorems
• On the essential spectrum of complete non-compact manifolds
• The Weyl Tensor of Gradient Ricci Solitons
• Gravity localization in a string-cigar braneworld
• Smooth metric measure spaces with non-negative curvature
• Classification of Nilsoliton metrics in dimension seven
• A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four
• Positively curved shrinking Ricci solitons are compact
• A renormalized Perelman-functional and a lower bound for the ADM-mass
• Bach-flat gradient steady Ricci solitons
• A note on the splitting theorem for the weighted measure
• Rotational symmetry of Ricci solitons in higher dimensions
• Rigidity Theorem for integral pinched shrinking Ricci solitons
• Para-Ricci-like Solitons on Almost Paracontact Almost Paracomplex Riemannian Manifolds
• Uniqueness of gradient Ricci solitons
• Gradient shrinking Ricci solitons of half harmonic Weyl curvature
• A family of 3d steady gradient solitons that are flying wings
• Structure at infinity for shrinking Ricci solitons
• Ricci solitons on low-dimensional generalized symmetric spaces
• Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime
• Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition
• Rigidity of asymptotically conical shrinking gradient Ricci solitons
• Rigidity and infinitesimal deformability of Ricci solitons
• Heat kernel on Ricci shrinkers
• Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers
• Reduction of gradient Ricci soliton equations
• Geometry of manifolds with densities
• On Potential Function of Gradient Steady Ricci Solitons
• Expanding solitons with non-negative curvature operator coming out of cones
• On gradient Ricci solitons
• The Curvature of Gradient Ricci Solitons
• Conical structure for shrinking Ricci solitons
• Optimal growth bounds for eigenfunctions
• Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature
• Bach-flat isotropic gradient Ricci solitons
• Splitting theorems on complete manifolds with Bakry-Émery curvature
• Three-dimensional Lorentzian homogeneous Ricci solitons
• The Euler Class from a General Connection, Relative to a Metric
• Uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous manifolds
• Time analyticity for heat equation on gradient shrinking Ricci solitons
• Singularities of Connection Ricci Flow and Ricci Harmonic Flow
• Remark on a diameter bound for complete manifolds with positive Bakry-Émery Ricci curvature
• Topology of Kahler Ricci solitons
• On curvature estimates for four-dimensional gradient Ricci solitons
• Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons
• A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons
• Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons
• Non-trivial $m$-quasi-Einstein metrics on simple Lie groups
• Harmonic Forms on Manifolds with Non-Negative Bakry-Émery-Ricci Curvature
• Compact 4-Dimensional Spin Gradient $m$-quasi-Einstein Manifolds Satisfy the Hitchin-Thorpe Inequality when $m\ge 1$
• A note on the uniqueness of quasi-Einstein metrics on ${\mathbb{H}}^n\times \mathbb{R}$
• Non-trivial $m$-quasi-Einstein metrics on quadratic Lie groups
• Curvature estimates for steady Ricci solitons
• Rigid properties of generalized $\tau$-quasi Ricci-harmonic metrics
• Cohomogeneity one shrinking Ricci solitons: an analytic and numerical study
• 2D viscoelastic equation from the perspective of Lie groups
• Gradient Ricci solitons carrying a closed conformal vector field
• From the Ricci flow evolution equation to vanishing theorems for Ricci solitons
• Uncertainty Principle and its rigidity on complete gradient shrinking Ricci solitons
• Rigidity of closed metric measure spaces with nonnegative curvature
• An upper diameter bound for compact Ricci solitons with applications to the Hitchin-Thorpe inequality
• Bounds on volume growth of geodesic balls for Einstein warped products
• Characterizations and integral formulae for generalized quasi-Einstein metrics
• Bach-flat noncompact steady quasi-Einstein manifolds
• Special solitons on 3-manifolds